Basically, what this comes down to is: If you want to have a real discussion with the other guy, you have to find out what he thinks. Arguing in favor of what you think will only produce a knee jerk reaction from him -- because he's never going to hear what you are really saying.
If you want to convince the opposition, start by finding out what members of their group say on your side. Both sides have discussions among themselves, members who don’t agree with the majority. They often have arguments that are couched in the terms that apply to the convictions of their party.
If you argue along these lines, you will be able to hold a civil conversation, even if you do not convince anybody. And you will come closer to having sympathy for your fellow man, even if he does not convince you.