I'm still learning to ride a motorcycle, and as I've written earlier - it helps knowing about countersteering. I don't care how or why it works, but when I push the right handlebar the bikes turn more to the right.
I've been riding bicycles for more than 40 years, and drove a cage for almost 30. I was not aware of the countersteering effect until my MC teacher told me a few weeks back, and when I got it I realized that's exactly what I've been doing on my pedal bike and why it helps on a moto as well.
Over here most MC teachers explain what countersteer is and why you might want to know it. I was fully able to swivel between cones without knowing it, but it turned out to be a lot easier when I got it explained :)