Their Relationships:

As we can see clearly, Rei & Asuka are opposite one another. Rei is quiet, Asuka is emotional. Rei hates red, Asuka hates dolls. That's why they're rarely in good moments together. Especially in the problem about Shinji. Rei likes Shinji and also Asuka. But Rei shows that she cares about him (just naturally, because Ikari Yui is Shinji's mother). In other way, Asuka deny her feelings to Shinji. She often angry to Shinji, insult him, etc. But like I've said in reasons section, Asuka just want to gave him advice because she thinks Shinji is so hopeless. OK, stop talking about Shinji. Asuka start to dislike Rei in episode 9 when Rei can make a better harmonization with Shinji. Asuka seems like jealous of Rei. Since that time Asuka always wants to be the best pilot or at least better than Rei. Rei? We won't understand about this first child's feelings. I guess she doesn't care about anything. But that makes Asuka dislike her more. Asuka is the kind of girl who hates to be delay. She prefer to be hated than to be delay. But you know what, Rei & Asuka's relationships is not so bad. We can see that Rei try to help Asuka in some battle (but then Asuka says, "Damn, that little bitch save me!"). And Asuka try to ask Rei to join with her, Shinji & Misaoto to eat steak at restaurant (but Rei is vegetarian). But well... they're just so different.