I do both. When I finish shaking the water out, I put it in the holder bristles up to allow the water vapor to escape upward.....I'm kind of paranoid about getting brush rot (if there is such a thing). The next time I use the bathroom I flip it back over. I really don't think there's a right or wrong way.

It's OCD with me. I spent $150 on my brush and want to get another 9 years out of it.