In a way, I'd say Everton are sort of our rivals too, only in a much more amicable way. They're the team we're most often likened to, it's the most contested fixture in all of English football, the banter with their fans is generally good, and we're in roughly the same situation as clubs right now.
It's not a rivalry in the traditional sense as I have a real soft spot for them, but it's one I enjoy more than most.
ETA: In my experience, their fans seemed genuinely sad to see us relegated and pleased to see us come back. I think they see the competition between us too.