It's not common, but it's dangerous. The technical term is suspension trauma. I climb solo quite often, and had to make my peace with the fact that if something comes down from above and knocks me unconscious, it could very likely be fatal.

I'm not sure it's accurate to say that hanging upside down is any better if a climber is incapacitated. As a practical matter, a saddle that was trying to continually flip you over would drive a climber crazy.

There's a bit of debate in the climbing community about the importance of suspension trauma, and how long you have before you need to worry about it during a rescue.