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Asuka tends to push everybody away, because she strives to become independent. She doesn't want to rely on anybody because doesn't want to be treated like a child. When she was a little girl, her mother replaced her for a doll (even calling the doll by the name of "Asuka"), and later her mother died. At that point Asuka cut-off all ties to childhood, striven to become the best so she could finally get the praise she never got from her mother.

When she first hears of Shinji, all she is told is how great he is, such as his advanced synchronization scores in the Sachiel fight while having no prior training. She views Shinji as an obstacle and as competition, so she tries to flaunt her skills and become better than him. But Shinji is far from a competitive person and doesn't react to her subtle call-outs; this frustrates Asuka because she feels as if he's not paying attention to her challenges, and he's still getting better scores without trying.

But slowly, she begins to grow feelings for him, feelings she herself can't even come to terms with. She tries to get him to react, to give some kind of signal to express how he feels about her. She tries reverse psychology in Ep 9 when she first begins to live with him, with the whole "Wall of Jericho" speech. In Ep 15, when they kiss, she gets upset that he didn't hold her.

Her feelings are finally expressed in EoE during pre-instrumentality. "If I can't have you all to myself, then I don't want anything to do with you", "You won't help me! You won't even hold me!", etc.

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Shinji also has feelings for Asuka. Her constant bullying sometimes makes him bite back, and gives him a back bone. He views Asuka as a source of strength, and late in the series, he turns to her for support. "Asuka... Please, wake up. Call me an idiot like you always do..", "I want to protect you and stay with you forever". Not to mention he smacked the monkey right over her comatose body.

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I believe that if they both just came clean with their feelings for each other, something could really work out and there wouldn't be so much tension between the two.

Asuka and Shinji definitely have feelings for each other, but they're both awful when it comes to expressing themselves. Both are drawn to each other, yet they also feel intimidation. It would all come down to whether or not Asuka would ever be willing to pull off her mask and let Shinji see her for what she really is: a damaged little girl begging for praise and acceptance. She's really a lot like Shinji, just filled with more rage and bitterness. I think Shinji would be able to sympathize with Asuka if he knew of her past and how it's the cause of her "independence" and arrogance.

In turn, Shinji would have to grow some back bone and if he's going to be with a girl that's arguably just as screwed up as he is.

For the record, RE-take made me into a huge Asuka/Shinji shipper. That said, I don't see it happening. Obviously things didn't play out well for the two in the anime, and from what I understand Asuka is an even BIGGER b***h in the manga and a lot more Rei/Shinji occurs. So the only possibility of getting to see the two in a cannon relationship would have to come from Rebuild.

Their relationship seems similar to how it was in the anime, but Asuka seems to be making an extra effort to win Shinji over this time around. Shinji also seems more capable, given how epic he was at the end of 2.0. But I'm not holding my breath. I'll be crossing my fingers just for Asuka to go through the next two movies without being impaled and disemboweled.