Jerkass Woobie

Normally, The Woobie is a sufficiently Moe character that:

• You'd want to sympathize with, sometimes to the point of wanting them broken to like them more.
• You'd like to hug or otherwise care for or help them.
• You want to see any villain who hurts them get hammered by Laser-Guided Karma.

The Jerkass Woobie produces a different visceral reaction: you'd like to punch them in the face yourself and you also want to hug them and say, "I'm so sorry that your life is such a mess." This character type is defined by three elements:

• First, the character must be a jerk. Even if they help people, they're not just selfish or insensitive - they're outright cruel, insufferable, immoral, amoral and/or the villain. They have a mean streak that tends toward the sadistic; they delight in hurting others, whether emotionally or physically, and often have little patience for anyone who disagrees with them. Also, keep in mind that sometimes, any sympathetic character does act like a jerk at some point, such as when a normal Woobie gets pushed too far, but this does not make them into this trope.
• They have a hidden, softer side. Many are clearly capable of being downright friendly, courteous, and kind, but either are too angry and impatient inside, or fear showing their softer side when not behind closed doors. Others are clearly Troubled, but Cute, with a Freudian Excuse.
• As if in return for their bad behavior, the author seems to delight in placing them in unpleasant situations, and yet continue to show that they are sympathetic. Basically, the universe hates them so we don't have to.

Examples

Ragna the Bloodedge from BlazBlue (pictured above). Though he does have his good sides and moments, he is for the most part rude, foul mouthed, verbally abusive and is perfectly willing to kill any and all innocent Punch Clock Villains that get in his way on his quest to bring down the NOL... In fact, it could be argued that the only thing keeping him from being an Anti-Villain Protagonist is the fact that 1) the real villains in the BlazBlue-verse are MUCH worse than Ragna could EVER be, and 2) the most active of the Big Bads have seemingly made Ragna his designated victim, and has put lots of effort into fucking Ragna's life up as badly as possible. For example by possessing the earlier mentioned Jin, Ragna's little brother, and, through Jin, cutting off Ragna's right arm.

Xenogears is chock full of them. Ramsus & Hammer stand out the most. Xenosaga continues the tradition with characters like the tortured former serial killer/victim of government brainwashing/cultist/spy Cherenkov & Albedo, whose tragic life as a Tyke Bomb and agonizing over the thought of outliving his brothers turned him into the Ax-Crazy Memetic Molester we all know and love. There's also Virgil, a first-class Jerkass who keeps harassing Shion, but who watched the woman who loved him and saved his life get eaten alive.

Michio Yuki from MW. Judging by his backstory of being kidnapped and molested by a criminal who becomes a priest in the future. Does that end for him? No, he gets exposed to the titular chemical warfare that not only turns him into the villain that he is, but also gives him occassional headaches, getting hospitalized at one time, and only has a few years left to live. You can't help but feel sorry for this guy even if he's beyond redemption.

Dr. Tenma from the 2003 version of Astro Boy. He's a control freak of a father to the point of abuse, yet watching the last few episodes one can't help but feel sorry for him.

Light Yagami from Death Note. He Used to Be a Sweet Kid, he was a Nice Guy with a bright future ahead of him before becoming a Serial Killer with a God Complex. Even the author says his life was ruined the moment he picked up the Death Note and over the course of the series he gets put through hell and fails and dies horribly in the end at the hands of Ryuk. But he has no one to blame but himself since not only his actions ruined lots of lives (including his own family), but they also brought huge consequences and were ultimately led by his desire to be a a sort-of God.

Spongebob Squarepants: Squidward Tentacles may be arrogant, rude, self-centered, egotistical and just downright mean, but he's also the Only Sane Man amongst various idiots, his Neutral Evil boss, Mr. Krabs, and Sandy Cheeks. Sometimes Plankton can be this too, mainly post-movie.

Demona in Gargoyles, both very evil and very sympathetic all at once.

Moral Orel: this trope is arguably one of the main internal rules of character writing for this show. No one questions that many characters in this show are repulsive people who do and say repulsive things, which is more often than not used to portray the hypocrisy of religious fundamentalism. However, special attention is paid to make it clear to the audience that disgusting people have feelings too.

Tom from Tom and Jerry. He is a mean cat that gets put through a lot of pain which is sometimes deserved and his own fault and sometimes happens because he's a huge Butt Monkey.