As an addendum to my previous post about being sick, looks like, as a guess, that I have “Vestibular neuronitis.”
The causes for this are many but it is commonly reported in people who have been sick with the cold or flu in weeks previous to it. I had a rather bad cold a couple of weeks ago so this fits. Basically, a virus or something has probably inflamed a key nerve in my ears, upsetting my balance and giving me mild vertigo. Radical head motion seems to make this worse, which means it probably is in my ears and not my central nervous system.
The power of the Intarweb!
In any case, barring bad luck or more serious causes, this usually goes away within a few weeks and gets better to a great degree after the first few days.
I probably shouldn’t try climbing on my roof until then though.