I'm pretty sure it's a weak or cracked solder joint somewhere on the mainboard. I yanked off the rear cover and tapped the mainboard lightly and the problem resolved itself. I could move it around a little and have it reappear and disappear randomly.
For the time being I jammed a paperback book underneath the board to support it fully and left the back of the TV off. Since I'm no soldering whiz and I don't want to drop several hundred bucks to get the thing repaired I'm hoping it'll last for a few more months until I can work up the nerve to buy a 32" LCD replacement. Every once it'll collapse vertically again and I'll have to wiggle the mainboard a bit to reset it.
Cheap factory soldering plus terrible design regarding the mainboard support (it naturally bows down the center with the cover on, with zero support there at all) plus lots of heat equals this problem I guess.