Adding to the previous discussions on counter-steering bicycles: you can see it in tyre tracks. e.g. ride through a puddle; or, ride from rain to under cover.
Consider: how do you steer riding no-hands? Shifting your body weight makes the front wheel turn (because of the "rake"), but now your bodyweight is on the outer side of the turn! Hold it for a (brief!) moment, then lean the bicycle the other way and you have a nice turn in the opposite direction - which is what you wanted all along.
This is a critical skill on motorcycles, and most people learn it subconsciously, but real masters can throw the bike into turns in ways that make no sense unless you take countersteering into account. I'd love to be good enough someday to make physics seem confusing...
As a cyclist and motorcyclist, I've always found that bicycles will let you get away with not knowing you're countersteering, but on motorcycles you really need to be explicit about "push harder to turn tighter".
Consider: how do you steer riding no-hands? Shifting your body weight makes the front wheel turn (because of the "rake"), but now your bodyweight is on the outer side of the turn! Hold it for a (brief!) moment, then lean the bicycle the other way and you have a nice turn in the opposite direction - which is what you wanted all along.