Also, there is no fixed correspondence between turning the wheel and leaning, because in this case you do move your body. Think about when you are in a car and someone turns sharply to the right. At first you lean left, but then you unconsciously right yourself.

Similarly on a bike, if you turn the wheel right, you will eventually either fall over, or turn to right (which will happen with an unconscious lean). Since the correction to lean-right is usually not done consciously, and happens very quickly most people don't realize there is an initial lean left.