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 talk with each other, you find that you're on exactly opposite sides of the fence on this issue. Compromise can'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 when the real trouble starts, because you two have a really hard time accepting your differences; instead, you fight about them. You think you're having a fair fight, but unfortunately, your partner is always the more powerful and more competitive of the two of you. You rely on your radical thinking to get you through an argument, but that is no match for your lover's implacable stance.