It only just occurred to me today that the swivel tray does not work. Not only that, it never *has* worked. I can turn the TV thirty degrees to the left, and it swivels right back to dead center. I can turn it thirty degrees to the right, and it does the same.
I know just enough about classical mechanics to get really really confused about how this is even possible. Maybe the TV is too heavy? The top surface of the entertainment center is very very slightly off of level; maybe that's the problem?
I'm confused -- but mainly, I'm annoyed with myself that I never thought to fix this problem before.
Anyway, I've fired off an email to the tray-makers to see if they have any ideas. If you have any ideas, o ye friendslist, comment away.
ETA: Aha! I went with apthorpe's suggestion of checking the center of mass. Specifically, I tried shoving the TV set backwards about an inch or two, and -- ta-da! -- everything suddenly worked. Hooray!
 ... but they claim the tray will support TVs of up to 250 lbs. My current TV is 68.3 lbs.
Two words: Duct tape. Seriously, if the problem is "not enough friction" then you can sure as heck introduce more friction into the system. Bubble gum would also probably work, as would various types of jams and jellies. Or maybe I'm mis-understanding the source of the problem? -J
Why do things rotate? Because there's an imbalance in forces. Nature is lazy and it only does enough to get by, at least in the time, size, and temperature scales we humans live in. Yes, superfluid helium will drain out the hole in the bottom of a cup, spread out along the outside bottom of the cup around the hole, and circulate up the side of the cup back to where it started. Bully for liquid helium - we're dealing with a TV and a Lazy Susan.
The pragmatic suggestion is the one previously given - make the swivel tray more difficult to turn. Friction or a wedge or something will balance out whatever force it trying to turn the TV in The Wrong Direction. The whole point of the swivel tray is probably to make the TV easy to rotate so as long as you can aim it in The Right Direction without pulling a muscle, you've added enough friction.
The less helpful suggestion is to guestimate the center of mass of the TV (actually the vertical axis of rotation through the center of mass - you couldn't care less about the vertical component of the center of mass; that only comes into play when the damn thing tips over on you.) Line up that axis of rotation with that of the swivel tray and that should reduce the tendency of the TV to rotate. Speaking as an engineer - good luck eyeballing that. Who knows how mass is distributed inside the chassis of your TV? Certainly not the guys who built it.
You might want to set the thing on a pair of toothpicks or something and move them around until it seems equally tilty. Mark where the toothpicks were with a pencil or something. Then move them under the TV to about 90 degrees from where they were and do that again. Contemplate the wonder of physics and experimentation while justifying that you're spending your precious limited time on this planet trying to find the center of mass of an infomercial delivery vehicle. "Soap - it's what's for dinner!"
So there are two complimentary tactics to use. First, center the TV on the swivel tray with the contemplative toothpick method above. If the TV still swivels, add friction until the damn thing stays put.
If things work out, please tip your Lazyweb Physicist generously... :)
PS: No, really. Liquid helium is seriously messed up.