I’m not even sure why I turned it on, but for some reason, some time in the past, I did turn on the Compiz effect called Freely Transformable Windows. I probably clicked on it randomly to see if anything dramatic happened, didn’t notice any changes and moved on, but if you look at the description it sounds pretty cool: you can rotate (in 3d) or tilt or scale any window, including its contents. If you look closer at the description though, you’ll also see that once the window has been “freely transformed,” you can no longer “interact normally” with it, meaning you can’t use it at all, or even close it normally — it’s like it’s locked up, with the window crooked.
So like I said, I forgot about selecting it, and had no idea what it did in the first place. Unfortunately, it has a bunch of Control-Shift-Letter commands to initiate the effect, which override other uses of those same keyboard shortcuts, so I would be in Firefox say, press Control-Shift-A for the add-ons page, and instead of getting what I want the window would go crooked and lock up. As far as I could tell, it was some weird and serious bug.
Well tonight I just happened to do a little Googling, and found the info that reminded me of what I did. I turned off the effect, and all is right with the world.