I ssh'd in and found that ARDAgent wasn't running. I tried the usual kickstart incantation:
$ sudo /System/Library/CoreServices/RemoteManagement/ARDAgent.app/Contents/Resources/kickstart -activate -configure -access -on -privs -all -restart -agent
Activated Remote Management
Stopped RFB Register MDNS
No dice. Stopped RFB Register MDNS? That's a new one on me. I ran ARDAgent by hand, and found that the login process had hung. ARDAgent doesn't restart until login completes, and login never completed. I had to killall loginwindow to kill that login attempt, at which point ARDAgent fired right back up.
I'm not afraid of ssh, and I'm a proficient command-line guy. I sorted this out fairly quickly. A quick troll through the relevant mailing lists/support sites, though, shows that in the Apple world, I'm apparently in the minority. And I don't expect that to change any time soon.