When I was a kid, I rode everyday that it was above nose-freezing - that is, when you inhale through your nose and you feel the hairs frost over. We kept our bridles in the house, so they were warm in the horse's mouth, and we'd ride bareback in the pasture. When we got cold, we'd just get off, take our bridles, and go back in.
Galloping bareback through snow is very interesting - you learn balance, and the horse learns lift. Riding in winter gloves is interesting, though, as you have little dexterity.
Later I learned that riding in snow can help teach a horse to lift his feet higher. The snow in this video might be a little deeper than most folks want to ride through though.
1. After changing the rules to allow riders to go topless, to increase spectator interest, dressage is now getting more television airtime than the AFL, cricket, tennis, swimming and golf combined.