Hats Allowed?

My new school allows teachers to choose their policies on two highly contentious teenage possessions: hats and cellphones. I feel like I need to give the administration a little shout out for giving teachers the space to decide what works in the classroom, but I don't know what to do about hats!

I was raised that men always took their hats off upon entering a building. Women wearing ball caps also took those off. This action communicated respect to the people around you, especially your host. It also showed professionalism.

One of my colleagues pointed out that disallowing hats for that reason is making a cultural value judgment. What if your home culture doesn't equate respect with removing your hat? I definitely agree with being as open to other cultures as possible in the classroom (with the caveat being that I think it's also my responsibility to teach how students will be expected to act at jobs and college). I also like being able to see a student's face at all times.

So, what would you do?