How do you remain friends with someone with views diametrically opposed to yours? Is it worth the effort?

I don't. I can be civil and have discussions with people, but if we don't share the same core values, listening to them on an ongoing basis is like nails on a chalkboard. I could be friends with someone who had different opinions, but it really depends on how deep down they go. Did we vote for a different candidate? Ok, no problem. Did they vote for a different candidate because they're a bigot? Big problem. We'll be waving at each other from a distance from that point on.

Hi Lago,

For your intriguing question, I tend to go for your option A – that friendship and camaraderie ARE more important than the opposing views…here is why:

When two people sit together and share that camaraderie, something deeper is going on; there is an exchange of some kind, a leveling, an opening if you will…and verbal discussion is not even involved.

And that may, indeed, be part of the reason Mr. Mike seeks you out, in addition to your good company, your very presence may just make him feel better. I think your friendship with him sounds lovely.
* * *
However, I also have a line that I draw; if someone is using me as a sounding board to get affirmation of their bigotry just by me listening to them and giving them an audience, then I courteously but firmly excuse myself.

Well for me I have different levels of friendship. I have good friends who we are very different in our beliefs but have sustained friendships for about 25 years. We have more to our friendship than just our core beliefs. We share other interests and enjoy our time together so we over look our differences. The difference is we respect each other deeply. 
I had an elderly Armenian neighbor, who one day asked me over for Armenian coffee. I love Armenian coffee in those little demitasse cups. So, I said yes. I went over and joined he and his wife for coffee. English is their second language. So,  we spent the hour gesturing, and pointing. They learned a bit more English, I enjoyed the company and their talk of their home in Armenia. We ended up starting a morning tradition that lasted for a few years until they moved out. Every morning I'd stop in before work and we would drink coffee and cigarettes. Soon his old cronies started coming around. We'd sit and I'd hear about their stories about their countries, their war wounds, getting older, their grandchildren. All in broken English. They heard my stories of my life up till that point. We had nothing in common, their beliefs and language was not something I understood or could relate to at that time. But we made due. We accepted our differences because we enjoyed our time together. Because in the end we were more than just our beliefs and our experiences. We were just being social beings enjoying our time together. I wouldn't trade that time for anything. Had I been stuck in our differences, I would have never given them a chance and I would have lost out on a wonderful experience. 

i probably wouldnt

Everyone's entitled to their own opinions on any subject no matter what they are within reason.......you can still be friends with them though.... This post was edited by Nice Jugs at March 29, 2017 2:56 AM MDT

I am, by nature, a master-debater - love the twists and turns of debate. However if someone turns hulk over it, and cannot engage in reasoned debate - conversation would probably be limited. I could get on with them tho and do.  There are some things I would struggle with, animal abuse, child abuse and abuse of women - I don't think I could be friends with those kinds - ditto I struggle with people who don't care about the environment.

Sometimes ya just gotta let people be themselves.

My neighbour for almost 30 years was a former WWII Luftwaffe mechanic. To him, Hitler was the greatest man of the century and the jews were responsible for the war. We had many discussions about it and, although we disagreed strong -- often loudly -- we managed to remain friendly neighbours. That's not the same as being friends, but I have no more right to impose my values on him than he does on me. Eventually he moved away.

I would rather have had that kind of workable relationship with him -- one in which we each knew where the other stood -- than an interminable standoff.