Hamlet is a classic example of this - is he a disturbed soul pretending to be crazy to trap the "sane", or is he honest-to-goodness bonkers? Modern interpretations generally agree that he isn't completely right in the head, but debate continues on how far this madness extends and on whether it resembles schizophrenia.
One alternate interpretation of Hamlet is that he is not an indecisive procrastinator. Rather, he is frustrated by the knowledge that, if he acts too soon, before he can prove Claudius' guilt to everyone else, his plan will fail.
In one of Hamlet's early soliloquies, he explicitly mentions that he doesn't know if "his father's ghost" is a lying apparition or not. He needs to prove Claudius' guilt to himself before he can act.
There's a critical essay on the play that points out that, assuming Hamlet was a Protestant (we know he studied at Wittenberg, which was a center of the Protestant religion in Shakespeare's day), he would have believed that good people go straight to heaven; therefore any ghosts must be the ghosts of damned souls.... in which case it would make sense not to trust them.
Another essay goes the other way around, claiming that, as a catholic (the play doesn't specify when the action takes place, it could be in medieval times), Hamlet would avoid killing Claudius before he knew for sure Claudius would go to hell. That's why he needed to be sure Claudius was the killer, and, after he was sure of it, he held his strike when Claudius was praying for forgiveness.
There's also some debate on just how much affection Hamlet had for his mother... and of what kind. Yes, some adaptations go there.
There is a minority suggesting that Hamlet and Rosencrantz had drunken fumblings with each other at Wittenberg. Well, how else do you explain Guildenstern acting so much like a guy who's been forced to hang out with his boyfriend's asshole ex?
Part of the problem is that Shakespeare never bothers to explain (and probably doesn't know) that the Danes from the original story he was cribbing from have a superstitious dread of killing the insane. Amlaeth's facade was a canny maneuver to buy time since his stepfather/uncle would kill him to secure his position. It is because of that dread that the bad guy tries to send Amlaeth to a foreign land to a monarch who didn't share that dread of killing the insane.
The play's script explicitly says that Hamlet's 30. Whether he acts his age is another matter (his immaturity might be part of the problem), as is whether the play treats him as an adult. (Some critics say that Shakespeare might've changed his mind about Hamlet's age and just forgot to take out that line).
The play's script as it exists today has contradictory information about his age. He's said to be in college; but Hamlet also says he knew Yorick, who died 30 years ago. Just shows that even in Shakespeare's time Writers Cannot Do Math. (Although this may also be a case of a reference to 13 being changed to 30 when the play was reprinted. Numbers got messed up in a few plays of the time because the original author's handwriting was illegible in some places.)
Or since Horatio is the only one left alive at the end, he is by necessity our Narrator, thus he would portray his friend Hamlet as more prescient and sensible than he really was. In fact, if Horatio wants to cast Hamlet in the best light possible, the interpretations of all the other characters in the play are suspect as well.
A theory bolstered by Hamlet's dying request to Horatio: "[R]eport me and my cause aright/ To the unsatisfied."
One that both neatly resolves the seeming inconsistencies in the story while opening up a plethora of possible new interpretations: the whole play is a Death Dream from the point of view of Hamlet's father.
The character of Claudius is full of ambiguities, and has wide scope for reinterpretation. Depending on the portrayal, he appears to be a competent, conscientious, well-liked king—arguably preferable, in fact, to his navel-gazing undergrad of a nephew. There's little reason to think that Hamlet would be up to the unpoetical demands of the job. In fact, it's only because of the Deus ex Machina that is Fortinbras that Hamlet's actions don't throw Denmark into rudderless chaos and invasion. There's "something rotten" for you.
Polonius' traditional interpretation as a doddering bore and fumbler might also be off. As a couple of modern portrayals have borne out, Polonius can be seen as a competent, occasionally ruthless official who understands more than many actors and directors give him credit for. His "neither a borrower nor a lender be" speech to Laertes—usually played for laughs—is perfectly sound advice from start to finish, and his opposition to his daughter's relationship with Hamlet proves horribly well-founded.
Additionally, readers, viewers, and Hamlet himself take Claudius' panicked reaction to Hamlet's staged play-within-a-play as open/shut confirmation of the king's guilt. However, Claudius' terror makes equal sense as the reaction of an innocent monarch whose crazy young nephew and heir has just staged a thinly-veiled portrayal of his impending death at the latter's hands.
Another is how much Claudius' love for Gertrude - if he truely loved her, like he said, shouldn't he have knocked the poisoned wine away from her? Or was he too far away and simply restrained himself when she started to drink (that depends on the adaptation)? Or did he just let logic stop him for one instant to continue his plan?
Furthermore, how much did Gertrude herself know — was she in some way complicit with her husband's death, or completely innocent of the whole thing? Also how does she react to the closet scene, in which Hamlet brings all kinds of accusations to her? Does she believe him and realise her guilt in betraying her husband by marrying Claudius, or insist that he is mad (which she later does in the graveyard, but maybe that was just to protect him)? IIRC, David Tennant's version had an interesting spin on it: Gertrude refuses to believe Hamlet, but when Claudius tells her not to drink the wine she suddenly realises that it is poisoned, Hamlet was telling the truth about everything, and then commits suicide out of guilt by drinking it deliberately.
Another thing that can be interpreted differently is the ghost: is he real, or is he a part of Hamlet's insanity. There is evidence for both theories in the play. The evidence that the ghost is real is the fact that Hamlet went to see the ghost after the others claimed to have seen it. Evidence that the ghost isn't real is the fact that there is a scene in which Hamlet sees the ghost, but his mother doesn't.
There's been at least one production where the actors playing Horatio and Fortinbras decided to imply that Horatio had secretly orchestrated the events of the play to clear Fortinbras's path to the throne.
Henry IV parts one and two: Is Falstaff a sad, old, cowardly man looking at the Crown Prince as a meal ticket, or is he an indefatigable mound of life and bombast who genuinely loves his Hal like a son? Is Hal a defiant, young hedonist living out the last flames of irresponsible youth before the weight of the crown comes crashing down on him, or is he a cold, calculating politician using the denizens of Eastcheap to craft his public image so he can appear so much finer in his reformation? Do Hal and Falstaff even really like each other? There are so many ways to play both characters. Compare the scene where Hal and Falstaff practice Hal's future encounter with his father the king: the 2012 BBC production and the 2010 Globe Theatre production.
Then moving on to Henry V: Is this truly the reformed Hal, the noble and victorious "star of England" the brave and courageous leader of a "band of brothers" in a just cause? Or is he still a cold master of his own public image, grinding up the "good yeomen whose limbs were made in England" to make an illegal and poorly thought-out land grab in France? In one breath he exhorts his soldiers, "on, on you noblest English!" and then in another threatens "heady murder, spoil and villainy" upon the civilians of a French town. His losses in the Battle of Agincourt are tiny compared with the French, but he did kill all his prisoners. Does he even know what he's doing at any given moment?
Macbeth: Was Lady Macbeth a manipulative and power-hungry bitch steering her weak-willed husband into regicide, or was she a feminist hero overcoming the role society had built for her and asserting her strength?
In the Agatha Christie novel Cat Among the Pigeons, another alternative is offered: Lady Macbeth was someone who liked the idea of being a power-hungry bitch willing to kill anyone who got in her way but wasn't really one: once she got a taste of what that was like, it drove her mad. Macbeth, on the other hand, thought of himself as the hero and needed a push to get started. Once he did, however, he found that he was in fact a power-hungry bastard willing to kill anyone who got in his way without remorse.
Then there's Macbeth himself and the witches. Did Macbeth kill Duncan because the witches prophesied he would become king? He admits that he had already been fantasizing about killing Duncan before he ever met the witches ("whose murder is yet but fantastical"), and that if their prophesy is true, he'll become king without killing Duncan ("then chance will crown me"). Has he perhaps already decided, subconsciously, to kill Duncan and seize the throne, and the witches have come to him to warn him? Likewise, does he kill Macduff's family and try to kill Macduff because the witches warned him to "beware Macduff," or had he already decided to do it; Macduff was already plotting against him, after all, which is why Macduff was in England when the assassins came.
In some productions the Third Murderer is Macbeth in disguise. In Roman Polanski's film adaptation the Third Murderer is Ross.
The character of Shylock in The Merchant of Venice is open to lots of Alternative Character Interpretations, largely because he's not drawn as unambiguously evil as other Jewish characters of the same time (Marlowe's The Jew of Malta, most notably.) Is he a greedy bastard who cares for nothing but money? Is he a hard-nosed business man who knows that his only protection from those who would like to see him ruined is his reputation as a bastard? Is he the victim of repeated bullying and abuse who finally gets what he believes is a chance to take revenge on the person who has abused him the most — and do it legally? The famous "hath not a Jew" speech in particular humanizes Shylock well beyond most Shakespearean villains, who tend to be For the Evulz or mad for power. Revenge killings were still common in Shakespeare's day, and still often considered legally justified. Shylock's also the only character in the play who doesn't try to twist the law or back out of bargains freely entered, despite these traits generally being ascribed to Jews centuries before and since, implying Shakespeare may have intended the character as a Deconstruction.
The character of Portia from the same play is one of Shakespeare's very best villains. After getting Shylock to give up his demand for the pound of flesh she, along with the Duke and Antonio twists the law around so much that Shylock has to give up all his wealth to the state and his ungrateful daughter (who, by the way, left his house by stealing his money), and also has to convert to Christianity, or die. Then she tricks her husband's ring from him, just so she can hold it over him as being "unfaithful."
Is Antonio in love with Bassanio, and essentially being asked to finance his own heartbreak? Is Bassanio a callous dunce cruelly taking advantage of his best friend's romantic feelings for him?
The character of Claudio in Much Ado About Nothing is consistently portrayed as a fresh-faced young ball of Wangst, whose willingness to be fooled by Don John and public rejection of Hero makes him look all the more callow and culpably gullible. If one takes Benedick's description seriously, though, Claudio is better seen as a huge, raging moose of a professional warrior, genuinely unused to female attentions, socially clumsy, and inspired to flights of language that aren't natural to him. He doesn't trust his good fortune; he's afraid of looking like an idiot, and half-prepared to hear that he's been taken for a fool all along. This combustible mixture doesn't make his meltdown and public treatment of Hero less horrifying, but it puts the character into focus and helps one understand why Hero might be prepared to forgive him.
An alternate interpretation of Iago's actions in Othello are those of a scorned lover. Iago manipulates the situation to destroy everyone around Othello except for Iago himself. If his plan had gone correctly the only person left that Othello could trust, and thus who he would naturally turn to, would be Iago.
Then there's the battle between people who believe Romeo and Juliet were really, deeply in love, and those who think they were just dangerously hormonal.
Romeo, a romantic hero who just found his One True Love — or dangerously passionate, fickle nutcase? Did he sweetly entice Juliet, or poison her reason with his vile tricks?
The Taming of the Shrew: No way Katherine's speech at the end about women needing to submit to their husbands is meant to be taken literally. She doesn't gradually weaken under Petrucchio's mind games but suddenly, out of sheer exasperation, grits her teeth and starts agreeing with him to get him to do what she wants. Who really "tames" the shrew here? Katherine seems to be the one who tames herself and learns self-control, which reaps more benefits than tactlessly bitching at the world.
The Musical version, Kiss Me Kate, calls for a sly Fourth Wall breaking wink at the end of this speech, a gesture that is no big deal now, but scandalous in the 40s. You can tell which side of the argument a director lies on by whether they keep or omit the wink.
This one goes hand in hand with the characterization of Petruchio. Is he trying to break Kate psychologically? Or is he trying to help her to become a better person through Tough Love? Is this a game for him, a contest for dominion, his idea of a friendly argument? One performance involved a Petruchio who was a good 30 years older than Kate (probably pretty historically accurate at that) which led to a paternal vibe of "This is going to hurt me more than it hurts you."
An alternate Alternate Character Interpretation: Petruchio has no interest at all at taming Kate (as he says early on, he actually likes her for her wit), but all his friends expect him to try to. Therefore he makes an absurdly big show of taming her to hint to her that he's not serious while he looks like he's trying really hard to his somewhat dumber friends. The scene where she comes over is where she finally gets it and starts to play along. The moral of the story, and especially the ending scenes, is not therefore "wives, submit to your husband" but rather "a happy marriage is built on trust and not the man and woman struggling against each other."
In most productions of Shakespeare's The Tempest, Caliban is a savage at heart, acting only on emotion and instinct, devoid of compassion; meanwhile Ariel is a pure creature of the spiritual realm, incapable of malice. However, some productions go as far as reversing the two — noting Prospero's vicious mistreatment of Caliban, they see him as a tragic creature, whose 'rape' of Miranda was a simplistic, disastrous attempt to express his love for her, for which he has been cruelly abused ever since; while Ariel is shown as an evil spirit who is kept in check by Prospero (why was he imprisoned in that tree in the first place, hm? Why does Prospero keep changing his mind about freeing him? Is it possible that a being who describes so poetically how he whipped up a massive storm and dashed a ship to pieces is a little dangerous?).
The possible interpretation that Ariel is actually gay. This was explored in the 1979 film version of the play.
It's stated right there in the text that Ariel was imprisoned because he refused to be a slave to the witch Sycorax, and the non-fatal destruction of the ship was done under the magical command of Prospero, so there's no evidence at all for Ariel being evil. The more common Alternative Character Interpretation is that Caliban and Ariel are symbols of colonial oppression, with Caliban being a rebel and Ariel being a collaborator.
A little more on Caliban: he has one line about how Miranda wasn't exactly refusing his affections. Brian Aldiss expanded this into a short story, with the two genuinely in love and Prospero forcing them apart for his own reasons.
Nora from Ibsen's A Dolls House is, in the conventional reading, a naive innocent who gets a harsh lesson in the world and grows up to move out. A wholly different interpretation is that we're actually seeing a plot involving serveral conspirators, organized by Nora, to break her tyrannical husband's hold. The Doctor being "ill" is a setup—he's fine and the card with the X through it is a signal he's ready for Nora to take off with him as they've obviously had a thing going on. Nils Krogstad's reasons for going along with the setup are obvious (knowing he's on the chopping block at work because of Torvald he has nothing to lose and everything to gain revenge-wise) and Nora is also, more kindly, masterminding the reunion of Krogstad with his old girlfriend and her old friend, Christine, who's been the victim of a similar domestic tyranny. Nora is in fact a Diabolical Mastermind orchestrating her husband's downfall.
Could there be a homosexual subtext to the relationship between Becket and Henry in the play Becket?
Brand by Ibsen: A fanatic jerkass who willingly lets his family die for his cause, or a tragic idealist who is doomed by circumstance? He has often been interpreted and shown as something close to the first, although Ibsen himself stated that Brand actually was meant to be a good guy.
In the musical 'Chess', opinions vary if Anatoly is a selfish jerk who betrays both his wife and his lover just to win a chess game, or if he's justified in choosing to put his own ambitions above the blackmail that the KGB and the CIA are trying to use on those he cares about. Or is he a man too proud not to give the game his best effort?
Florence; desperately lonely woman with horrible taste in men, likely stemming from her Daddy Issues, or selfish player who managed to make both of the world's best chess players into her pawns, despite Freddie's dodgy mental health and Anatoly being a married man with kids?
Freddie - Anatoly wonders this aloud and we never know one way or another; is the American completely out of his mind, or is it a carefully cultivated gambit to bring him wealth, fame, and publicity, as well as unnerving the hell out of his opponents?
Anatoly's wife: A poor woman hoplessly devoted to her philandering jackass of a husband, or someone who puts up with his shagging other women, abandoment of their kids, and betrayal of their country because she gets the prestiege, wealth, and perks of being married to a national hero?
The Arbiter: someone only in it for the love of the game and completely above all bribes and manipulation, or a representative of neutral countries who are mutally pissed off at the Americans' and Russians' mutual posturing and gleefully enjoying the chance to tell them both to go screw themselves?
Siegfried from Der Ring des Nibelungen. Is he an Ubermensch who refuses to follow the corrupt lawful society? Or a complete Jerkass?
Is Wotan a Well-Intentioned Extremist trying to create a lawful world and committing suicide when he realises his time is up? Or a self-serving figure who commits suicide from despair?
There are various readings of Marlowe's Doctor Faustus. There have been productions that managed to turn Mephistopheles into The Woobie. But even if you don't go that far, whether Faust is a Tragic Hero who is frustrated by a callous god's refusal to do anything and honestly wants to improve the world, or a bored academic who was mostly just trying to summon demons for the lulz and had no idea what he was getting into, or a Villain Protagonist who doesn't care about the rest of the world at all and is only in it to enrich himself is open to personal interpretation.
In Molière's Don Juan, the character of Don Juan's servant Sganarelle is obviously not the sharpest knife in the drawer, but, given the nature of plays, it's possible to portray him as decently crafty, given his intellect and low education. That is, you put a bit of subtext on his seemingly stupid comments in certain scenes. So it's possible he's not a complete buffoon — until a recent English translation added a line that can't possibly be read as anything less than comic relief.
Jesus Christ Superstar reinterpreted the character of Judas, stating his fear that Jesus was no longer in control of the situation as his reason for betraying him, rather than simple greed. Although one might find Judas quite a sympathetic character in the musical, he can, however, also be seen as a coward looking after number one.
The most recent DVD version, starring Jerome Pradon as Judas, seems to suggest that Judas was romantically in love with Jesus, to the point of obsession, and there's a scene where Judas appears jealous about the attention Jesus gives to Mary Magdalene. Therefore his later actions in betraying Jesus could be seen as those of a lover scorned.
Another interesting interpretation is that the entire play is happening in Seymour's head. This theory is supported by some scenes, most noticeably "Suppertime" when Audrey II is singing quite loudly, but Mushnik is unable to hear it.
There is also the theory of the Villainous Tenor in some Gilbert & Sullivan works, particularly The Mikado which holds that Nanki-Poo is in fact selfish and shirking his princely duties to the point of lining multiple innocent people up to be killed to cover his escape, rather than merely being a victim of Katisha's machinations. Perhaps more credibly it has been advanced for Frederic in Pirates Of Penzance, questioning whether he is a slave to duty or merely going along with what seems his best chance of survival, first going straight as piracy isn't working out for the gang, then joining the pirates again when they have a gun to his head and then refusing to switch sides even when they capture his apparent love interest (who, he wooed largely as an easy way of going straight).
Recent revivals of Oklahoma! have switched the roles of Curly as the "good guy" and Jud as the "villain". They present Curly as a self-absorbed Jerkass, while Jud is a lonesome, hardworking everyman who dreams about things he cannot have. In these productions, by the end, rather than thinking "Jud got what he deserved", you think "That asshole Curly got away with murder!"
A 2012 production at Seattle's 5th Avenue cast a black actor as Jud, adding "racist douchebag" to the above insult against Curly. In fact, every other character in the show could be seen as racist based on the way they treat Jud.
Even his attempt to kill Curly with the trick eyeglass isn't fully unprovoked or self-serving. Before that, Curly just randomly shows up at Jud's house, starts up a song about what it will be like when Jud is dead, and when Jud gets pissed off at this, Curly gives what is really an underhanded threat of killing him by showing off how good a shot he is, inside Jud's own house, no less.
On the other hand, there's Jud's "Bartlet farm" speech, where he recounts how a hired hand burned down his employer's house (killing the entire family inside) because the farmer's daughter rejected him. Jud claims to have gotten the story from another hired hand, but some productions have implied Jud himself was the killer.
Raoul: Foppish pretty boy who was not interested in Christine until she became famous, or romantic young man fighting for his love?
Carlotta: Cartoonish diva villain, or Technician being forced out in favor of someone less talented, coincidentally right after she disappeared overnight with the theater's wealthy patron?
Madame Giry is one of the more enigmatic characters, too - talk to ten different fans and you will probably get ten different motivations behind her actions.
The sequel Love Never Dies (2010) has made this even worse: Raoul has been turned from a brave and noble suitor into an angry alcoholic and gets blamed for his failed marriage to Christine because he's just not good enough for her. Critics of the show take the view that Raoul is the real victim, having been driven to dissolution by ten years of living with an emotionally distant and faithless wife and an irritatingly precocious kid. As well, in the 25th Anniversary staging of Phantom in 2011, Raoul seems to spend half the time exasperated with Christine and half the time bravely and boyishly adoring her, which may be a hint of what's to come in the sequel. Especially since Andrew Lloyd Webber cast his two Love Never Dies leads in the concert, it seems like he's deliberately trying to make a point that wasn't there. Fans who ignore the existence of Love Never Dies just think it's a more layered way to play Raoul, a role that can easily come off as dull or one-dimensional.
Johanna can be seen as the token Broadway Love Interest, as she apparently fell in love with Anthony within the space of two seconds; or she could just be using Anthony to get an escape route away from Turpin's clutches.
Another interpretation of Johanna in the stage version is that she's something of a budding Sweeney herself. This is based on her abusive childhood, possible inherited mental instability, the posited manipulation of Anthony, her time in the asylum, and her willingness to shoot the asylum-keeper (in the stage musical; the film has the asylum-keeper killed by his "children").
Justified due to availability of talent: Toby can be cast either as a child or as a mentally retarded man. The only description given is that he's "simple." Thus, The Movie has a Toby the size of your thumb, whereas stage productions often cast a man decades older (though the 1980s taping has a Toby that you can accept easily as a man playing young — until you notice the mustache).
Neil Patrick Harris in the 2001 concert recording may be an example of an adult Toby — although with Johanna and Anthony being played by the definitely not teenaged Lisa Vroman and Davis Gaines, he could be playing younger as well.
Another interpretation of Toby, based on the ending of the stage version where Toby is shown as a grown man locked and straightjacketed in an insane asylum, is that it is an adult Toby telling the story from his own childhood memories and that the adult actor is used consistently to ensure that the audience knows that he is the same character.
Though not as major as the other characters, the Beadle certainly counts as well - he can be seen as either a complete psychopath who's no better than Judge Turpin, or a Well-Intentioned Extremist police officer who believes a Hanging Judge is the only way to enforce the law.
The wonderful Avaric. Virtually no one accepts him as the carriage driver he is in the musical, he's either the jerk like he is in the book or... well anything from a serial rapist, murderer and blackmailer to a better love interest to Elphaba than Fiyero and it's not unknown for one fanfic author to express both views of him in different (or the same) fics.
Glinda near the end. The implications of her more then platonic friendship for Elphaba, and her general personality invoke this in both the musical and books. Her being a Stepford Smiler is the most common, though fans tend to dig deeper.