Swivel varies by model. It's really a matter of weather the stand lets you tilt it 90 degrees. One easy way around this though is to just mount it on the wall. You can get the VESA mounts for ~$25 on eBay. Plus, if you're going to have it around your computer anyways, it'll save you the space by having it off your desk. Just check to make sure the monitor is VESA compliant. I don't think I've seen any current monitors that aren't, so it's not much to worry about.
EDIT: Buro, did you get the 21" FP from Dell? I think I recall someone mentioning over on the shmup boards that they switched a bolt around or something on the stand of that monitor to make it swivel the other way.
But yeah, a lot of recent verts on PS2 have both viewing modes, left or right... not sure if I recall Ikaruga having it though (at least on DC). The Cave ports do though (I think). So does MAME.