Like a liberal and a conservative or a religious zealot and an atheist, you two take two opposite sides of every important issue that comes up. It's even likely that there's a central issue that is important to each of you -- something that has shaped your life and molded your viewpoint -- and when you start to talk with each other, you find that you're on exactly opposite sides of the fence on this issue. Compromise just doesn't bridge the gap.
You're just too different. There's no chance for balance, because you're on two ends of a very long seesaw. Vegans and butchers; cops and criminals; couch potatoes and world travelers... You just do things and see the world in two completely different ways, and ne'er the twain shall meet. That's where the real trouble starts, because you aren't able to accept your differences; instead, you fight about them. You think you're having a fair fight, but unfortunately, you are always the more powerful and more competitive of the two of you. Your partner relies on their radical thinking to get them through an argument, but that is no match for your implacable stance.