I am with handles it don't matter. Although I do the opposite rear up. Main reason is I skin my animal to allow cooling of carcass ASAP. First off the hams need to be seperate to allow adequate cooling of the inner hams, the use of a spreader bar to hang the animal will alow this. Secondly when I skin I leave the head attached to the cape for transportation and identification(unless a mounting buck) - for this reason alone skinning rear down is much easier to me. I always wash the inner cavity with clean water and prior split brisket/pelvis. To remove hairs from the skinning process I simply wipe down the carcass with water and a dash of vinegar.

Again like you how I was taught, my dad being a butcher said what you put in is what you get out of your meat in terms of quality.