I can ride pretty well without hands as well. When I heard about counter-steering a long time back I paid closer attention to what I was doing while riding hands-free. Turns out that I was doing a weird little bodily flick motion in to get the wheel pointing to the opposite direction of the turn before I started leaning into the turn.

Pay attention to yourself next time you ride and you might be surprised.

Yep. If you try to just lean right immediately, the bike will just.... fall over, because the wheel stayed straight.

The momentum from the flick your body creates begins to turn the wheel a small amount, and the forward momentum of the bike in combination with the lean keeps the wheel turned right. By contrast, if you flicked the bike/wheel but did not lean, the wheel would simply straighten itself again.

Even for automobiles this stuff is very simple suspension + steering mechanics. Due to how their suspension & steering components are designed, and assuming they are aligned properly, a car will always follow the road. If the road leans + curves, so does their steering.

Yes, I discovered my weird hip tilt when I was using a bad seat. It was then that I realised I barely used my hands to steer at all (sharp turns excepted)

There's a really cool video demonstrating the idea [0]. They took a set of handlebars and welded them to the frame. You'll see that you can affect a motorcycle, but only poorly. On a bicycle, you're much more top heavy and can force the vehicle to capsize far faster. On my motorcycle though? If I lightly increase the throttle it'll go absolutely straight (which is nice if you need to reseat your gloves, but scary if the road starts to veer to the side!)

[0] https://www.youtube.com/watch?v=4PbmXxwKbmA

You can, I think both were described in the theory book for notorcycle license when I glanced through it.

Fun fact: when I read about it I had to test and I had to make sure I wasn't deceiving myself so I put one palm in front of the handle on one side and the other one behind the handle on the other side. That way I could ensure I pushed it the correct way.

It worked exactly as advertised (50cc bike so somewhere below 60km/h).

Much simpler than using slow motion camera etc.