The main reasons for it are the very unusual ways the characters save the world, and the fact that the abundant religious symbols' religious meanings and Evangelion meanings are totally different - the creator even admits that they were only added for flavour.
It involves a giant robot being transformed into the tree of life by a set of nine other giant robots crucifying themselves and a huge tuning fork, then getting plunged into the vagina-forehead of a giant alien who has taken on the form of the male lead's mother's clone. This is to save the world. Probably.
Then everyone gets a hug and turns into glowy Tang, there's some conversation about human nature, the giant alien falls apart and spurts a giant stream of blood onto the moon, and lastly, "Kimochi warui". Post any segment of End of Evangelion at or past Lilith's rise for other people to watch and a good portion of reviews will ask "What the *** did I just watch!?!"
Disclaimer: Unless it's Jet Alone, in EVA "robot" means "giant cyborg made by cloning an angel, growing it in a probe, putting it into armor that restrains it to a 80 meters tall vaguely humanoid form, then transplanting a woman's soul into it" and getting her child to pilot it.
If you think Eva is confusing, watch RahXephon and cry your brains out. Over half of the series is spent Contemplating Our Navels and the backstory is revealed almost completely without exposition; it's so hardcore that it's nigh-impossible to piece it all together. Special mention goes to figuring out all the connections between Ayato and the rest of the cast. The Movie is clearer but not by much.
The Black★Rock Shooter OVA. Hands up, everyone, if you went "What was THAT?" when it ended.
The 2012 anime version goes into more detail, and more confusion. Everyone has psychotic "other selves" in a parallel universe who are trying to kill each other. Why they're so violent, why they even exist, and so on isn't explained very well. The whole thing with the picture book about a rainbow-colored bird was supposed to be an explanation (linking colors to the "other selves?"), but it's too vague to help much.
The movie of Ghost in the Shell is cramped with obscure symbols, random quotes, and lots of philosophical discussions that come out of nowhere and are over as suddenly as they came. The second movie is nothing but 96 minutes of obscure symbols, random quotes, and philosophical discussions, from the beginning of the opening to the end of the credits. And this is Mamoru Oshii, after all.
FLCL. It may take you two or three viewings to understand just the plot, which is, in and of itself, just a Coming of Age story.Fridge Brilliance is everywhere and the show will only make sense after multiple viewings. This comes as a surprise to many since everyone is too surprised that there's actually a plot. The manga escalates this by being basically its own story, yet still requiring knowledge of the show to try to follow it.
To elaborate on this a bit, the first four episodes of the story are (relatively, mind you) coherent. The remaining nine are so absurd and dadaist that it makes Evangelion look completely self-explanatory by comparison.
Uzumaki starts getting weird within the first 10 pages. Then things are constantly getting weirder, and it's not until the very last page that things are starting to make sense again.
The Big O, though it wasn't entirely intentional: the planned third season was apparently supposed to explain things in a more straightforward manner, it's just that they didn't get to.
Revolutionary Girl Utena. The anime is screwy enough, but the movie rivals End of Evangelion. Utena's director Ikuhara has expressed particular, almost sadistic, delight in the despair fans have shown over figuring things out. Some of his more famous replies to fans have been "Miki keeps timing things because his watch contains the secret of the universe" and "The reason Utena turns into a car in the movie is because I really wanted to turn a cute girl into a car."
Dead Leaves takes this Up to Eleven. It begins with a guy who looks like Canti from FLCL and a girl with a weird eye marking waking up naked in the middle of nowhere, and ends with a super-intelligent (?) baby coming out of the girl's panties with Guns Akimbo, putting enough dakka in the air to kill a bull elephant, and flying off into space to kill a giant worm. Retro, the Canti Expy, frequently comments along the lines of "This is so screwed up."
Relatedly, the director Satoshi Kon has been much kinder than most and explained the movie as "following dream logic" and that we should just go with it.
Ergo Proxy: Lain with shotguns. Full of mind-screw situations - especially when Proxy One starts playing mind-games with Vincent Law, his host and when the identity of Real herself comes into doubt later on. The group also has other weird experiences, like an episode focusing on the characters taking part in a game-show, and an episode where Pino explores a Disney-like theater complete with anthropomorphic animals. When compared to the rest of the tone of the series, it's no wonder most of the cast are near-crazed by the end.
Tsubasa Reservoir Chronicle. So ambiguous and convoluted at times that official translators admit in the notes that they're making a guess and winging it. Unofficial translators do not say such in the notes only because that takes up space better filled by actively and profusely cursing at CLAMP. Making a timeline for this series is practicallyimpossible. And don't even think about trying to create a character chart. Even the characters are baffled by some of what's going on. Among the characters that have expressed a great deal of confusion is The ChessmasterBig Bad. AKA, the man who planned a decent chunk of the confusing events. And believe or not, in a Crowning Moment of Funny, now that Xxx HO Li C has ended, even Word of God has admitted that they too are rather confused over how everything turned out and want to re-read it.
While it didn't get it nearly as bad, poor sister series Holic definitely got its moments when it got dragged it into Tsubasa's mess.
Geno Cyber, because there's an unexplained Time Skip between the first and second arcs, compounded by the fact that the first arc has No Ending.
Suzumiya Haruhi. The Anachronic Order will mess you up, since you cannot relate half of the stuff being said. Furthermore, it gets hard to follow in the 4th novel Disappearance and its continuance in another novel, where time travelling is combined with alternate universes. It may take you 2 times to read just to understand how they actually managed to solve it. The 9th (current) novel features two realities, for no apparent reason, in which different stuff happens. And it ends with a Cliff Hanger, so we still don't know what's going on. To be fair, the more surreal of the two phone calls in the 9th novel is obviously made by someone who knows more than they really should, and as it's the first point of difference, it's likely the cause of the split. Who they are and what they want is unclear.
And in the 10th novel it solved out to be that there were two realities, one of which a backup, and the mysterious sixth member of the club is Haruhi's subconscious. Yes, you didn't misread this, no need to reread the sentence.
The time where we enter an endless recursion of time. Reading/watching Endless Eight for the first time will have anyone trying to figure out what in the world is actually going on.
Earth Maiden Arjuna. On one hand, the general theme of letting the earth be and saving the environment from unnatural influences is pretty clear. On the other hand, the details of the plot are rather surreal, to say nothing of the presentation...
Chaos;Head is one right from the beginning. The viewer is forced to pay attention to everything that goes on, not knowing if it's simply a delusion or possibly real. Even when things get cleared up, there's still another layer of mystery beneath that.
Being the spiritual successor to Chaos;Head, Steins;Gate proves to be a massive Mind Screw, but inverts it because everything is solved or explained by the end.
Alien Nine. Even if you read the entire manga and watch the anime more than three times, all keeping the coming-of-age metaphors in mind, you are still most likely to be left scratching your head at something.
Tekkon Kinkreet starts out pretty straightforwardly. The ending, on the other hand...
Haibane Renmei is a much gentler version of this; while the plot itself isn't all that ridiculous, the backstory and setting are never explained.
It, Lain, and Texhnolyze all have one thing in common: Yoshitoshi ABe. Thankfully, this is probably his gentlest work. Most of the mind screwing is in the nature of little mysteries left unexplained to everyone... including the characters, making it less of a shock and more of a curiosity that accentuates the surreal setting.
The art style alone is trippy (and beautiful) but the stories are explained in the end. It's heavily rooted in Japanese mythology, though, so the "answers" are even more freaking bizarre than the original "questions" ever were.
You have Azumanga Daioh, a slice-of-life anime. The concept seems normal. Then they bring in a giant, floating cat that hates the color red and is voiced by Norio freakin' Wakamoto! Though Chiyo-chichi only exists in one character's dreams/daydreams. The other characters are appropriately weirded out when she mentions it in real life, besides Osaka, who saw him once as well, after Sakaki already made some hints about it. She is just screwed up enough to put everything together, but she did give Chiyo-chan the stuffed animal.
Night on the Galactic Railroad is a rare example of when all the mind screw, that has been thrown at you during the movie, is... mostly explained at the end.
Ouran High School Host Club has hints of this, though you'd have to look really, really closely to notice them. Things like showing a shot of a character looking out of an empty window, turning the camera away for a moment, then turning back to the window, only this time the pink-tinted clock tower seen throughout the series is visible through it. Hey, it's made by the same guys as Revolutionary Girl Utena, what did you expect?
In the first episode, clips of light bulbs turning on are shown as more of the characters realize Haruhi's secret. In the last episode, during an interaction between Éclair Tonnerre and Haruhi, a shot of an unlit light bulb is shown. (Which then turns on once she discovers Haruhi's secret as well. Problem solved.)
The fun starts in Season 3, when suddenly the main enemy is a severely psychologically damaged Human Alien intent on proving their affection by putting the protagonist through horrendous trauma. That's the straightforward part.
A mild version at the end of the Lucky Star OVA, to give a brief synopsis; everyone turns into frogs and Shiraishi flies around singing. The scene was possibly another Shout-Out to Endof Evangelion as well as the Urusei Yatsura movie Beautiful Dreamer (a legendary Mind Screw itself); Shiraishi is actually wearing one of the character's outfits.
Darker Than Black has a lot of weird Techno Babble concerning the Gate, and some fairly critical things are never really explained. The last episode of the first season gets special points, given that it includes a segment inspired by Evangelion's Gainax Ending (specifically the "Congratulations" scene). The second season has even more of a "reach for the aspirin" ending. And another Eva Shout-Out for good measure.
Casshern Sins is kind of hard to understand in general, but you can tend to take all the symbolism in stride. Episode 18, however, just kicks your brains out. It seems a fairly standard episode of Lyuze, who is trying to kill the robot who killed her sister, but actually loves him too. She dreams of her sister telling her not to forget her. But in the middle of all that, we have random photographs of a woman. No, I mean... PHOTOGRAPHS. As in of a human, not an anime character. Given the themes of the episode, it may be her voice actor, but who knows?
Yakuza Girl. So much. Starts off as a standard tits-n-goreseinen sort of manga, and just spirals into "WHAT IS THIS I DON'T EVEN" territory. When a psychic preventing the bombing of Hiroshima by freezing the plane in time (and midair) for 50+ years isn't even the strangest part of the series, you know you've got a Mind Screw series on your hands...
Cat Soup is one extreme 30-minute instance of this trope.
Key the Metal Idol, commonly described as the strangest anime ever made when it was released.
Urasawa's 20th Century Boys is mostly just confusing because of overlapping crazy gambits and... stuff... but it's also stuffed with symbolism everywhere, all the time, and when the 'flashback' scenes start to overlap with the 'present time' scenes through a virtual reality simulation of the main characters' neighborhood in the Showa era created by the Big Bad, which various characters enter to find out what really happened, and in which one character's consciousness survives for some time after being shot in the head, it has crossed the line. Especially since you have to read the companion series Twenty-First Century Boys just to find out the ending.
Excel Saga is mind screw incarnate. Particularly the anime. The entire series is built on this premise (no joke). However, if you presume that the entire series is a mind screw, and then look at it in the context of being a mind screw, the specific mind screw episodes start to make sense as contributing to a plot and setting which is mind screw which are in turn metaphors for the mind screw symbolism of a post-modern society... Wait...
Fullmetal Alchemist relies heavily on Hermetic symbolism. The conversation between Truth and Father during the last chapter clearly shows that reason of Father's downfall was actually misinterpretation of Hermetic philosophy and stealing other people's power instead of self-development.
Bobobo-bo Bo-bobo, in which a man with a blond afro with extra-long nose hairs goes around trying to stop an empire based on the premise of destroying all hair in the world. Joining him is a... THING that looks like the sun, and a living block of jelly. These people have abilities called the Fist, and their abilities are stored in a personal MacGuffin called a Hair Ball, and you die if you lose it. Forget trying to explain it. It's not meant to make sense. Bobobo-bo Bobo-bo isn't a Mind Screw; it's just flat-out set in Cloudcuckooland.
Digimon Tamers. It may be a kids series, but since the main writer (who is not new to Mind Screw) relies on realistic dialog, often just with quick comments explaining what's happening, often the meaning behind the dialog is lost by the audience. One needs to really pay attention or rewatch it several times to appreciate it.
He's also a fan of HP Lovecraft. However, his use of Magic A Is Magic A means nothing outside the known rules of what can happen ever does, and the surprises are never Ass Pull. (If anything, it's more grounded in reality). We find out how the Digimon began, and just how denizens of Cyberspace can pop up in the real world. Craziness that coexists with a well-told, coherent story with characters that manage to keep acting like themselves. Tamers will never spout nonsense at you in hopes that you'll figure "I have no idea what the hell they're talking about, so it must be too profound for me", while remaining just as freaky and horrific as those that do.
Gregory Horror Show can and will leave you wondering what just happened in the span of a 2-minute long episode. And not just for the viewer, either—in the first two series, the main character (whose eyes are you seeing the action through) doesn't understand what's going on or why, and in the third series we find that not even the lord of the freaky place is immune to it.
The King of Thorn movie. You know an anime is a mind screw when the latter parts of it completely invalidate what we were shown in the earlier part.
After School Nightmare is rather hard to explain. At first it seems like a normal manga with an intersexed protagonist... then it quickly changes into a psychological manga full of Freudian influence, similar to Evangelion.
Jewelpet Kira Deco, oh boy. Let's just say from the beginning that the 4th series is a big WTF mind screw from the beginning. It did stated it's a general magical girl anime for girls with lots of sugar, but some crack slipped into the sugar in the form of Kazuyuki Fudeyasu and Makoto Moriwaki. Everything in the series doesn't make perfect sense. Why? Because the premiere we got a lot of heavily decorated things, a bus with jewel decorations, the moon turned into a television and it gets out of hand as all kinds of insane stuff appears in every episode that is Getting a lot of Serious Crap Past the Radar.
Milk Closet, by the guy who made Alien Nine. It's about kids who can jump across universes. The first chapter is confusing enough, and it just gets weirder from there. It ends with the three main girls fusing together and giving birth to an entire universe.
Murasakiiro No Qualia is absolutely full of this. From philosophy to quantum mechanics, the series has a field day with it. Perception is troublesome, after all.
In Naruto, the fact that Haku's a boy was a strange mind screw, and one that persists to this day, considering at least half of the audience still thinks he's a girl. Haku is subject to Rule 63 so much that him being genderbent is more common than the reverse.
One filler episode has Naruto's Shadow Clones rise up and usurp him. It's hard to make sense but it goes something like this: the clones are resting on a ship when something drops on the real Naruto's head. A sleeping Naruto clone wakes up and takes the hit. The dream the clone was having when he died mingled with Naruto's thoughts and made him hallucinate. Still keeping up? The real kicker is when the hallucination ends with Naruto being killed because they accused him of being a clone. Are you mindscrewed yet?
The OVA Ciel in Wonderland for Black Butler. Since it is another take on Lewis Carroll's Alice in Wonderland, this shouldn't be a surprise. However, it is surpriseing how dark and serious it gets after all of the weird imagery and characters we see.
Lupin III sometimes drifts into this territory in the anime.
Many fans have been confused about what the order of events was, or who the winner was, in Green vs. Red.
Lupin III: The Woman Called Fujiko Mine used a number of Mind Screw elements, especially where the series was preparing to reveal why Fujiko acted the way she did. "Because she's Fujiko" is basically the answer fans were given, the entire story arc leading up to the reveal getting turned into a Take That, Audience! for getting invested on her tortured history as a little girl.
Yuri Kuma Arashi, by the same creator as Utena and Penguindrum above, is about bears who can disguise themselves as humans. While this concept, and a few others, are explained at the very beginning of the series, a large amount of details regarding the setting have to be learned through observation - and, just like the creator's other works, said setting is a World of Symbolism. Good luck!