Heh, I personally think that Asuka actually does hate Shinji, and quite profoundly so. I'm not sure that she loves him; attraction is easy to prove by going over the series, but anything more is imho squarely a matter of opinion. For Shinji this seems easier. It's easy to see why he would come to lover her - he openly admires a lot of her attributes, and it is just is his nature to cling to people. Whether you call that deep attachment love or not depends entirely on how you see love. So, this question seems to be a personal matter entirely, and there are no "wrong" interpretations here. Note also that before the DC edits, their relationship lacked a lot of the romantic overtones, which are pretty ambiguous in either case, so not seeing any is entirely reasonable.
So, why does she hate him? Imho, the most deep-seated source of that hatred is the subconscious recognition that he is everything she despises in herself: he is weakwilled, meek, dependent, shy, self-deprecating - in short, he is the antithesis to how she tries to be. So of course she hates him, because he is everything she hates herself for, because deep down, she is just like him - or at least she fears that she is like him. And given what we see in the series, that seems to be substantiated. Long before Shinji ever gets dragged down so low, she runs away from home and tries to commit suicide by starving herself, because she lost her ability to pilot (and was forced to confront all her inner demons by Arael in a pretty hostile fashion). Without her Eva, she completely shatters. Much like Shinji, who completely breaks after losing the last person he could (maybe hope to) depend on: Kaworu. The only difference is that at their lowest points, only Shinji was put in the position to decide the fate of the world. If their roles were flipped, we would have seen Asuka going through the same journey he took, and she probably would have ended the world just as he did. Anyway, this is getting into fanfiction territory.
She has of course other reasons to hate him, as well. He abandoned her when she needed him most, he never took the initiative, he never answered her advances, and he never even looked at her and seen the real Asuka. Him masturbating to her is such an egregious offense to her because he didn't see her as a person - he used her as an object (to fuel his own self-hatred by making himself irredeemable in his own eyes; "Look, I behave like a monster, so I am right to hate myself, and now I can hate myself even more!"). He tries to do it again in Instrumentality, and again she refuses to be an object for him to cling to.
Which is highly ironic, because what she wants deep down is somebody who can take care of her, so that she doesn't have to. Because she is, after all, just a lonely child who wants somebody to stave of the evils of the the world for her. Somebody like Kaji: an adult who won't abandon her and take care of her. Even though she says she wants to be alone, she hates being alone. She says she wants to be independent, but she hates being independent: it means that if she makes a mistake, she has only herself to blame - take all the responsibility, get all the blame. If she fails, it means that she is a failure - it makes her worthless. She blames herself for failing to connect to her Eva. It's all about her - she wasn't strong enough, even though she should be. It's of course impossible for her to live on her own, just like it is for Shinji, because they don't have the infinite strength required to do so. Shinji fails to live up to the challenge of breaking through her wall and become the strong person she can depend upon, which she might percieve as his biggest betrayal
To bring this back to the question of how they could become friends: well, that depends on how you see things. After Instrumentality, they might know all these things about each other. At the very least, they know all these things about themselves. If they are a little perceptive, they will notice all their similarities, their similar experiences. Will that make them hate each other more or less? Will Asuka be more forgiving towards Shinji when she recognizes that he was rotting away in his own metaphorical bathtub when she needed his help during her final fight, or will she hate him even more? I honestly don't know. I can't help but feel empathy for people who have lived through experiences similar to my own, and I think that is a very powerful connection they share. But I listed a lot of things that might be hard to forgive above, and given her final words, I can totally see her just being done with Shinji completely after he has shown that he hasn't changed at all by reverting back to his utterly weak self. She at least managed to show kindness to him by caressing his face. So her rationalizing things to herself in a manner of "I changed for the better, he changed for the worse, so just fuck him" seems entirely plausible to me. To go into fanfiction once more: Who knows if that is permanent. 10 or 20 years down the line, they might make contact again.