Hang 'em high
An old butcher explained to me that cows are always butchered head down because the valves in the veins allow the blood to drain back toward the heart (and thus out of most of the body) when the critter is head down. Yes, veins do have anti-backflow valves.
Makes sense to me. I hang a deer that way when I can to butcher it.
As to field dressing them, that oughta be done at the site of the kill--don't haul all those guts back to camp! Just more to haul, then you have to get rid of 'em besides. You field dress a deer with its head uphill or up on a stump, so's the innards roll out as you release 'em. (Why do you suppose it's called FIELD dressing??)
So: Briefly: Field dress with head up, in the field. Butcher with head down, in camp.