My wrist is constant as well. As Lisa said, it makes sense to me that you use the same motion for the 2-hand and 1-hand swing. So no wrist turn.
With cleans and snatches it is common to see people angle the bell on the ground and grab it with the thumb turned in. This makes the clean and snatch motion a bit smoother and lessens the banging on the wrists. On the decent they'll twist the wrist just prior to the snatch motion.
There are a number of good drills to teach swing mechanics. The "towel swing" is one I like because the swing quickly becomes self-correcting.
(I should add that different kettlebell schools teach different approaches to the swing, snatch, etc. So your mileage may vary. Reif and the examples I used are focused on the RKC approach)