The following circuit is a MOSFET driver.

When I feed a 20KHz squarewave in, I get 20KHz out ... but as a
1.25MHz ring. This is as unloaded.

http://users.tpg.com.au/hburns/index.html

I assume this has to do with the air core coupling, but would like to
know more about how this happens. IOW what factors control the
presence and frequency of the ringing.

Can anyone please explain?

Jim Starret