Correct way to swap over dressage whip?

[livetoride]

It also astonishes me how few riders can bring a short whip from the carry position into action(reversing it in their hand).

Yes, very true. When I was taught to ride back in the dark ages, we were required by our instructor to practice upending the whip 100 times with each hand, every evening until we could do it without thinking. I still almost always use a short whip like this. Far more effective than a backhanded tap.