We live in Manitoba, so unless someone lives in Nunavut, it doesn't usually get much colder then the temps we endure We ride literally all winter. I couldn't stand the idea of not being around ponies just because of a little wind chill. We have no arena, just the front yard and open fields. We wear many layers and the horses are ridden all winter. Bareback is almost a must though, it keeps you WAY warmer when it gets frigid.
Our horses live outside, so we just make it a point to never get them sweaty. Other then that, we do all the normal things we do in summer, save for jumping of course.