Most vets I’ve seen will use both positions to do heath checks, just for the easy access to certain parts that both give. Something like gender checking & spinal alignment is always easier with 4 paws on the ground & a lift from the base of the tail, but for things like teeth, legs & paws, on the back is easier. I’ve got one on baytril & metacam at the minute, a burrito & quick flip onto her back is the only way to get it into her, she hates it but she’s not good with being handled in general.
As said it doesn’t hurt them, but for a prey animal it’s not a position they enjoy.
The unsafest way of handling them that can do damage, is lifting/carrying//gripping them by their ribs with no support elsewhere for their weight, because they have what our vet calls “floating” rib cages. Not technically floating, but the front of the ribs aren’t connected to a large sternum for support & rigid like ours are. Theirs is very small with tiny muscles & cartilage connecting them that make the cage more flexible in its movement. Coupled with thin, delicate bones that are easily broken, too much weight resting on them or pressure from fingers can easily do damage either to the connections or bones.
