Who knew there were so many thoughts and opinions on this subject? Here are mine in no particular order:

• I took my visor off because when I was in the drops I had to crane my neck back to see down the road. I put it back on when I ride my mountain bike on a sunny day. Why? Because it helps keep the sun out of my eyes? Because I'm a dork who thinks mountain bike riders need visors though road riders don't? Because I paid for the thing and I'm darn well going to use it? Probably all of the above.
• I always put sunscreen on the back of my neck so I don't worry about the sun. Stop to turn it around when the sun's hitting the back of my neck? I don't think so.
• I don't see how a bicycling cap would fit under my helmet without the bill getting in the way. Flip it up? How?
• I wear a mountain helmet all the time. When I checked out helmets it was the one that had the most coverage on the back of my head. I have a big melon and helmets usually sit on top like a mushroom cap. (See avatar. That isn't my regular helmet.) This actually goes down a ways in back. It's probably a little heavier than a road helmet but my neck hasn't seemed to mind.