One email and one thirty minute phone call (where the operator said that I might have "knocked something loose" when moving the console) later, and I'm now waiting for UPS to deliver me an empty box, in which I will place my poor 360 and send it off to the legendary Xbox repair center somewhere deep in the wilds of texas.
Now, I'm not a certified Xbox repair person, but I've pretty much narrowed down the problem to an overheating issue — if I crank up the A/C the 360 will power up and play Worms for a good five minutes before crashing and refusing to reboot. Maybe my heat sink melted off? I hope they have some of the non-melting ones in stock down in TX.