When I run into a conflict of seemingly "opposite" ideas, I often seek a higher principle which can help me incorporate what seems to make sense in both. Sometimes I will stay invested in one of the sides, making a modified argument, but I have a tendency to embrace a "third way" which I see as an improvement over both (keeping in mind that there are usually other "third ways" out there.) Sometimes this means that both sides respect what I'm saying, and sometimes it means what I'm saying is generally despised, because people insist that I either join with them or their enemies. It can be a practical necessity to join one side or another, but this doesn't effect how true anyone's views are. This isn't just about interacting with other people, the whole process goes on internally as well (at least I think it should.)
Even if I've embraced a view which separates me from both sides in a conflict, I still respect people who maintain one of the positions, because their confirmation bias points them in a different direction than mine does. This means that they will notice things that I will miss, because they are seeking out different kinds of information. Thinking that conflicts can usually be resolved by facts alone is something that doesn't sit well with me because of this. Thinking about the perspective you use to explain these facts is also important. Even if, despite what I've just said, we find ourselves viewing an entire perspective as nonsense, one question is still worth asking. Why does this person think this way? In thinking about that, we can find that maybe he or she wasn't as wrong as we had initially thought. We should keep in mind that our own understanding is always limited, and wrong in some respect or another. There is a wider picture which no one can ever fully understand. Other people's understanding is a tool for expanding our own.