That's the idea behind the Rain Design i360, a turntable that encloses the base of an iMac or Apple Cinema Display, and then allows easy movement of the display. Want to show a co-worker on the "back" side of your iMac something? Just spin the entire computer around. Need to plug or unplug something? No problem -- just swing the iMac or Cinema Display around and do it.
I've had an i360 on my first-generation Intel iMac since just after I bought it, and I can't count the number of times that it has saved me from having to pick up the computer and slide it around to plug something in. I've recommended the i360 to many of my Mac consulting customers who have iMacs or Cinema Displays, and many of them end up buying the device.
The i360 base has a bit of weight to it so it doesn't move, and there are rubber feet on the bottom to insure that your desktop isn't scratched. The computer or display base is held in place with a single screw, so installation is simple. The manufacturer's teardrop logo cutout (seen above) has a clear plastic window, so you can put a picture or logo into it just for fun.
There are now two models of the i360. The original, available for US$39.90, works with the 20" and 21.5" aluminum iMacs, the 15", 17" and 20" "plastic" G5 and Intel iMacs, and the 20" and 23" Cinema Display. To match the iMac you own, the base of the i360 comes in either painted white or in bare aluminum. The new model (US$44.90), called the i360 24", fits the 24" and 27" aluminum iMacs and the 24" Cinema Display. Alas, there is no i360 for the 30" Cinema Display.
The i360 has served me well over the years, and it surprised me to find that TUAW hadn't reported on this cool piece of hardware before. If you find your iMac sitting in the wrong direction often, it might be worth getting one and taking it for a spin.
Apple iMac 27-inch with Retina 5K display (late 2014)