Firefly from Ace Combat The Equestrian War. There are times when she acts like an a-hole towards other ponies, especially Fluttershy at the beginning (going as far as to smack her!) as well as Derpy and Medley later on. But deep inside, she's just a troubled mare, trying to readjust herself to the circumstances around her. Thankfully, RainbowDash is there to help her.
Brainbent features a few, which isn't surprising since the majority of the cast are dealing with some emotional problem or other, many of them fairly serious. Karkat is probably their reigning champion. He's a shouty little ball of rage and self-loathing, coated in a complete lack of tact, but he is, at heart, a compassionate person. When he isn't buried over his head in his own issues he's genuinely concerned for the people around him, and for all that he can be painfully inept at dealing with his own emotions, sometimes he gives good advice.
Naruto is one of these. He's quite willing to consider the feelings of others, but his experience with the Kyuushingai has made him rather caustic.
Sakura proves to be one, surprisingly enough. She might quarrel with Ino frequently, look down on people of lower social standing than herself, and lord her queen-bee status over everyone else...but when Aoi savagely beats Ino during Sasuke's tournament, Sakura shows a considerable amount of compassion because, as she herself declares, nobody deserves to undergo something like that.
Suigetsu may be a complete and utter prick to Naruto, but he likewise takes Aoi's beat-down of Ino very badly.
Suigetsu: Hey, I might not be the nicest guy around, but even I have some standards.
Aramus could be argued as in The Legacy Of The Blood Ravens. He's incredibly arrogant, and doesn't hesitate to charge into battle or lead his men into danger, but he'd give his life for the Chapter and in the end, wants what's best for the Blood Ravens as a whole.
Tama in The Lion King Adventures. Despite being the Big Bad of Series Two, she actually saves Simba from dying of a terminal illness in Sick Simba. She changes her ways in Goodbye, Nala, finally becoming an ally.
Mortality: Gregerson may be a jerk, but like Watson, he seems to care about Holmes.
In Mega Man: Defender of the Human Race, even though he's a bad guy until the episode 11 finale and is not fond of most humans, ProtoMan has demonstrated acts of kindness, especially to Nomad, Mega Man and Roll.
Shatterstorm and Roaring Yawn both show their jerkish sides when they first appear in Equestrylvania. However, the moment something traumatic happens, they end up coming to the defense of the ponies they've belittled. In Shatterstorm's case, he's shown to actually be pretty good with kids.
Despite acting like a total jackass most of the time- make that all of the time- Chaos from the Tamers Forever Series still has some moments where he shows that he genuinely respects his other half, and that he can actually be a nice guy when he wants to.
Ryo also fits the bill, or as Takato puts it:
he knew Ryo was just trying to be a friend… a crazy, narcissist, extremely annoying friend, but a friend, after all.
Zao, the Headmaster of the Imperial Naval Academy in The Odyssey. His initial address to the assembled class consists of telling them that they're all going to die horrible deaths in service of the Realm. Despite this, he really does mean well.
In Sonic the Hedgehog fanfiction "Prison Island Break", Shadow shows that under all his Axe Crazy perverted psychopathy, he's a very deep person. Since they're all in prison, he tries to conceal his softer side. However he privately acknowledges that Sonic is a better leader, and has become incredibly loyal to the team, even willing to risk his life to protect others.
Narration:What Shadow the Hedgehog said, and what he thought, were not exactly always in the same carpool. On the flip side, they changed lanes so often that they'd clipped each other's wing mirrors off. Still, he certainly wasn't a hero. And nobody could make the mistake of calling him a good man. But he was quick-minded, and fastidious, and determined, and clean, which are all becoming of a hero. The sole quality he lacked was virtue, and he knew it. Still, good men who believe their virtue to be satisfactory have already taken a step down the ladder, and bad men who think themselves sinful enough are ready to take a step up.
L becomes this in Story of the Century. Or if he'd always had shades of this it makes itself more apparent by the end of the story before he dies. Deconstructed in that his default "Jerkass" behavior pushes people away and he refuses to let on whatever capacity for kindness that he has.
Arthur's friend Kay from The Student Prince is elitist, sexist and just plain obnoxious, but when he's offered money by a reporter in exchange for details on Arthur's private life, he responds by punching the reporter and smashing his camera.
Kay: I'll stick your bloody camera where the sun don't shine, you scum! Offering me bloody money for dirt on Arthur! You're not worth the mud under his shoes, you leeches, even if he is a bloody poof! He's our bloody poof!
In Kill la Kill AU, we have Nui, being the "asshole sibling" who liked to bully Ryuuko when the two were three and two years old, but, however, at the same time, she does care about her, as can be seen in comic 27 when she makes it clear she doesn't want her to be sent away to a remedial boarding school. Likewise, there is an eight-year old Ryuuko, who is shown to care about her sisters and was distraught when she thought she was going to be sent away.
Arlene Johnson, Annie's boss in the Criminal Minds fanfic To live again. She's a classic Mean Boss and wants to use Annie's connections with the FBI for personal benefits but also knows how to motivate her people and wants the best for the station and it's cops. She's also shown to geniunely care about Annie (especially later on the story).
Pretty much every single member of Eric's crew in Sonic X: Dark Chaos follows this trope. They often fight, make fun of each other, and make cruel jokes, but they're all firmly on Sonic's side and are willing to help others even at great cost. Sonya in particular embodies this trope - she's highly arrogant and willful, but also has a strong sense of justice.
The rewrite makes Tsali of all people into this. Under all his genocidal Axe Crazy psychopathy, he has quite a few Hidden Depths. He considers his Tsali Endoskeletons as his children and shows remorse that most of them are gone. Likewise, when he hired a group of mercenaries to help him hunt down the Seedrians, he became quite fond of them and was utterly devastated when they betrayed him - it actually causes him to consider whether getting revenge was really worth it in the end. The fact that one of them was his little sister probably didn't help.