I think that I fixed it. I re-connected the 4 pin section of the ATX cable, while leaving the 20 pin section alone, and for some reason it now turns on and off as it should. I hope that it doesn't decide to do something else.
One would think that, but I plugged and replugged it so many times that it couldn't have just been a loose connection...unless the cable had a broken wire in it, but if so I can't reproduce the problem now.