She's sometimes seen as the archetypal tsundere, her (perceived) opposition to Rei paving the road to a thousand standoffs between tsundere and kuudere, such as Nagato vs Haruhi; but it doesn't seem right to me. Firstly because Nadia, from Anno's previous anime was much more of a tsundere, but also Asuka never felt like a tsundere to me in the first place; and I think I just found the words to explain why. She definitely has the "tsun"; she is hostile to the point of aggression towards Shinji, and it would also seems that she might be interested in him romantically (the "dere"). However, she lacks a causal relation between the two. See, the idea of a tsundere is that her hostility serves to hide the romantic feeling towards its object, which she doesn't want to admit. Asuna, however, isn't bashful about love. She has no qualms hitting on Kaji or kissing Shinji. The cause of her hostility is another feeling she represses and that's seeing herself in Shinji. The need she feels to be strong, talented, perfect, and getting praised for it is a huge part of her character; which is why she does away with any doubt or weakness. But Shinji is a reflection of all these weaknesses, and he doesn't have the "decency" to hide it like she does, and that's what makes him unbearable to her. And yet, perhaps some part of her wishes she could let herself be weak like he does. Like the widely hated line "What, don't to kiss a girl on the day of your mother's death?", she's making fun of supposed mommy issues Shinji might have, but really making light of her own mommy issues to help her dominate them. I think this plays intoher aggressive tendencies much more than romantic feelings, hence why I find the term "tsundere" less than appropriate.
No two unrelated people of the opposite sex can spend that much time in close proximity to each other, working as close together as literally playing Twister and sharing clothes, and not feel absolutely nothing. They'd both have to be robots, or be like Rei.
Additionally, Asuka's response to her classmates calling her and Shinji newlyweds is pretty telling that both of them have to be at least interested. Asuka shows jealousy towards Rei and Shinjis existing relationship, though it is basically devoid of romance entirely. While she always tries to take the spotlight, it is important to note that often she is competing for Shinjis attention and/or praise.
I definitely think Asuka has romantic feelings for Shinji, and it can act as a motivator for her to pick on Shinji. People often pick on others that they have an interest in, for various reasons. But if someone feels nothing for someone else, they usually don't pick on them. Usually they do nothing, because they have no feelings for them and therefore that person isn't even on their radar of potential targets.