Brief recap: when you hit the flush lever, the water level in the bowl would rise a bit, and then it would just kind of hang there, circling lazily for an approximate eternity. Any solids in the bowl would not go anywhere, but revolve a little.
It most assuredly wasn't the drain to the sewer line, because if that were the case the bowl would overflow. This was a problem between the tank and the bowl.
Thinking that an in-tank sanitizer cake had somehow gotten sucked down the water supply tube, I took the tank off...and discovered that no, there wasn't anything clogging that passage. Having established this fact, I figured that the water passages in the bowl were clogged. That probably meant I'd have to buy a new toilet, because the alternative was to buy about a gallon of hydrochloric acid from somewhere, find a way to plug up the toilet, and fill it full of HCl.
This evening, though, I realized something: the flush action depends on a jet of water flowing fairly quickly towards the drain. (Filling the bowl full of water will do it, but only if you dump in a lot of water, fast.) This is called a siphon for obvious reasons, and if that siphon passage got clogged with gunk the toilet might do the kind of thing it was doing. I reasoned that it would take perhaps five minutes to test my theory. The coat hanger that I'd used to try to unclog the main water passage was even still in the bathroom, and sure enough I discovered that perhaps 1/3 of the passage at most was open...and that was after I'd worked at it with the coat hanger for a while.
Needing another, more robust tool, I hit the garage and tried to find a piece of metal rod that could be bent to the right shape, and failed to find anything. Then I remembered the slotted screwdriver I've got that's been bent to shit for years; I straightened it, mostly, then clamped it in the vise and bent a 90° angle into it.
Thus equipped, I returned to the master bath and started scraping the crud out of the siphon. The stuff coming out looked like sand, and once I'd broken it up it flowed quite naturally to the bottom of the bowl. There was a lot of it.
Then I withdrew it from the bowl and hit the flush lever. Mirabile visu the damned thing flushed like it's supposed to. A bit slow, but I think it's always been like that--it's not a "low flush" toilet, so it doesn't need the velocity that low-flush toilets do.
The bowl is amazingly crummy, though. I tried using regular-strength toilet cleaner and was able to get the worst of the blue sanitizer crud off, but the lime/calcium deposits are bad enough that they don't mind ordinary toilet bowl cleaner. I got the leftover CLR from the Jeep heater core flushing project (I'll need a new jug for that job anyway) and dumped that in, and will let it sit overnight. Tomorrow, then, I can try cleaning the toilet again and see if it's any better. I might buy another bottle of CLR specifically to dump into the toilet bowl, in its entirety, and let it sit overnight again tomorrow, just to get the rest of the crud out of the water passages.
It'd be very nice if this was fixed, because quite frankly I don't want to be buying a new toilet right now. As it is, the thing needs a new seat and might need a new flapper valve, but if that's all I need to buy, I'll take it!