This is when a character can go from evil to good and still retain the characteristics they had as a villain. There are plenty of reasons for why his immoral or unpleasant behavior remains the same despite changing his alignment:
The reformed character may still have issues with one or more of the members of the group whom he initially opposed, particularly The Hero.
Sesshomaru started as a villain trying to kill his own half-brother InuYasha, but turned into his protector due to taking a young orphan girl under his wing. This doesn't mean he really enjoys showing kindness to others, even Inuyasha.
Much like the game continuity, Shadow in Sonic X pulled a Heel-Face Turn by sacrificing himself to save the world from the Space Colony ARK. In the Metarex saga, however, Shadow returns (with his memories erased), still the Perpetual Frowner, still cold and aloof around others, and still ruthless in his goals, even to the point of fighting Sonic and his friends if it's necessary. Believe it or not, he's largely more ruthless and less personable than his game incarnation (who mellowed out a bit more after his own title).
Exeter from Scion was as a menacing minor villain, but becomes The Big Guy of the main group who is less merciful than Ethan might like.
Emma Frost from X-Men started as a villainess who is cold, aloof, arrogant, sociopathic, and likes to underdress. Even after she joins the team, nothing has changed about her attitude and behavior, although she seems to let her coldness melt a little.
To an extent, Nao from Perfection Is Overrated, who remains bitter and cynical compared to the others even after her Heel-Face Turn, although she no longer believes that hurting others in revenge is justified or eases her pain.
The film adaption of How the Grinch Stole Christmas! has the main character learning the true meaning of Christmas. Even so, he still behaves like a jerk, but his heart is still a half-size too small, so that's good enough for everyone.
Catwoman again in The Dark Knight Rises. Her Heel-Face Turn ultimately comes about by her expanding on her good traits, and Batman encouraging her to consider others more (though she always had a soft spot for the little guy). But even when she has fully completed her Heel-Face Turn and fights alongside Batman, she's still more than willing to use lethal force against lethal force, as opposed to Batman's Thou Shalt Not Killand is the one to plug Bane with the Batpod's cannons.
Accelerator in A Certain Magical Index killed ten thousand teenaged girls, but transitions into The Atoner. He will do good to others, but he'll still behave like a utter jerk in doing so.
In A Clockwork Orange, Billy-Boy (novel)/Georgie (movie) and Dim become policemen after Alex is released from prison and proceed to abuse him when he encounters them.
As described on Nightside, Razor Eddie, "Punk God of the Straight Razor", is a reformed — but not retired — Serial Killer, who now targets those who threaten children and the homeless. He's been known to leave buildings soaked in blood but conspicuously absent of corpses; the narration calls him "an extremely disturbing agent of good. The forces of good didn't get a say in the matter".
Severus Snape from Harry Potter. Eternally loyal to Dumbledore after his Heel-Face Turn, but still a moody Jerk Ass (and Stern Teacher to an extreme degree), more than willing to punish Harry for the sins of his father and take revenge on his enemies (even gleefully allowing one of his old bullies to be taken off by the Dementors to have his soul removed!), and blatantly biased against Gryffindor, but still on the good side.
Angel: Faith post Heel-Face Turn is very much a darker version of Buffy, particularly in this spinoff, where to the series’ end she's immoral, rough 'n tumble and aggressive. It would be exaggerated earlier in the comics before she takes a big dose of maturity, and is now regarded In-Universe as better than the main character.
Buffy the Vampire Slayer: Spike started off as a villain in season 2, but by season 4 (partially due to a dose of Badass Decay) he sometimes worked with the Scoobies and he pretty much joined them during Season 5, while still keeping his mean streak as a vampire.
Farscape: Bialar Crais was the Starter Villain for the first season, chasing after the heroes until Scorpious came and take his place. While he underwent a Heel-Face Turn and backslid a couple of times, he never seems to lose his nasty streak.
This is quite common in the professional wrestling world. A good number of wrestlers begun their careers as heels, but were so loved by the fans that they just have to turn face despite still behaving the same way they started. A noticeable example is "Stone Cold" Steve Austin during the rise of his popularity.
Joshua Graham from Fallout: New Vegas actually had a Heel-Faith Turn after becoming one of the most ruthless and cruel men in the Legion. Even though he now fights for a noble cause, he still retains his ferocity and Kill 'em All mentality.
Granted, Proto Man's defining trait is that he's an aloof loner. His few skirmishes with Mega Man after 3 are more a test of his brother's mettle and resolve. Curiously, Mega Man 4 implies that Proto Man, despite learning of Wily's true nature and saving his brother at the end of the previous game, was still working for Wily up to that point (albeit as a Reverse Mole) due to Wily's reaction to Proto Man saving Kalinka.
Sagat from Street Fighter aligns himself with Shadaloo after suffering a loss to Ryu in the first SF tournament. He realizes that his revenge-driven desire in defeating Ryu isn't the right path, so in Street Fighter Alpha 3, he reforms from his ways, but still retains a heated yet healthy rivalry with Ryu.
Various characters of the Sonic the Hedgehog series started off as enemies of Sonic, but become either reluctant allies or just still clash with him on occasions when their goals or personalities clash against each other.
Brainy Twilight has recently begun an attempt to reform a cannibal pony. Progress is... iffy.
My Little Pony: Friendship Is Magic: Discord, antagonist of the season 2 opener "The Return of Harmony", is reformed by Fluttershy at the end of the season 3 episode "Keep Calm and Flutter On", but clearly demonstrates in several episodes of season 4 (such as "Princess Twilight Sparkle" and "Three's a Crowd") that he's still as big a Troll as ever.
Another example is Princess Luna in the Season 2 episode "Luna Eclipsed". Princess Luna still speaks in her booming voice early on, left over from her time at Nightmare Moon. This, combined with her past reputation as Nightmare Moon, makes every other pony scared of Luna.
Trixie as well, at the end of "Magic Duel". Although she gives a heartfelt and very sincere apology, she is still as hammy and conceited as ever. Just more apologetic, that's all.
Trixie: Don't you think the Great and Apologetic Trixie is the most magnificent and humble pony you have ever seen? Twilight Sparkle: *Rolls eyes, then smiles a little*
David Xanatos from Gargoyles arguably never had a Heel-Face Turn; he simply stopped being an enemy to the Gargoyles. (Granted, he was grateful they saved his son.)
Angelica Pickles, in All Grown Up!, is much less antagonistic and considerably nicer than she was in Rugrats, even if still prone to mischievous and selfish behavior.
Kuzco from The Emperor's New Groove is a lesser example. In the movie, he's a selfish, self-centered jerk who plots to bulldoze the entire village for his own profit and almost left Pacha for dead, but had a change of heart and becomes good friends with Pacha. In The Emperor's New School, however, he still behaves like a jerk and his heart of gold doesn't show very often.
In the series, his selfishness isn't as bad as he was in the movie, and it was out of his stupidity most of the time.