You'll try to cooperate with each other, but instead, somehow, you'll get into power struggles again and again. You will try, consciously or subconsciously, to force your friend to change, because their methods of processing the world and their place in it won't match up with yours. You'll probably push them to take a stand on their principles, and to defend their overblown ideas.
You'll both take your own side in a serious and dogmatic way, clinging to your own beliefs so fervently that you could forget there's any other way of experiencing the world. In some big ways, you'll disagree, but instead of trying to listen to each other and respect your differences, you'll put each other on the spot and try to force each other to defend your position. Challenges like this can be really hard to deal with, and could make you feel divided as a friends. On the other hand, this relationship could push you to define yourself in ways that you've never thought of before.