I usually carry my whip in my right hand. When in a dressage ring, there usually is no high rail for the whip to hit. The only real disdvantage is when doing a circle and the whip is on the outside. I have to flip the whip a bit harder to go across my thigh and still hit the haunch (since the bend puts the haunch a bit further away. If a horse has a really bad left side, then I will carry in just the left hand.
I never change the whip to the other hand in a test....never. The inconsistency in contact that it would cause is not an option, IMO.
There will be times when you will be wanting to correct the weaker side of the horse and if you are carrying ONLY ONE whip then it is more advantages to carry it on the side you need it the most.
I ALWAYS ( outside of competition) carry TWO whips. This reduces the rein inconsistency contact and I can correct either side right away.
As far as the dressage judge seeing the whip mls posted that is not correct as half the time one side of the horse will be seen by the judge and half the time they will not. So the whip is carried in the hand the rider feels it will be the most useful and comfortable for the rider/horse.
The only rule here is that you must salute with the NON whip hand.