The couple of times I have been on fairground wheels they have gone part ride one way and then changed direction.
One wheel that does go in only one direction is the London Eye. Looking at it front on, i.e. from across the Thames, it travels in a clockwise direction. Due to it slow rotation, and as you don't actually sit down and can walk about in the capsule, the motion is barely perceptible.
By the way, this isn't a variation on the Friday stupid thread is it?