Tweetbot!
Okay I can't honestly blame Tweetbot, but that's the app that gummed up slide over on my iPad. I was reading an article about how great the new Tweetbot is, and I realized that I hadn't really taken the time to peruse it. Back out to springboard, and I see that the Tweetbot icon is dimmed, as if it's in the middle of an update. But it's not. And not only that, I can't delete it, even after a restart. So I log out of my Apple Store account, and that unsticks it. As I'm redownloading Tweetbot it occurs to me that that's the last app I tried in slide over.
Now, with all my apps unstuck, slide over is working as it should. I still don’t know how my apps decided to get stuck in the first place, but let it be known that a stuck app may very well take slide over down with it. And, as we suspected above, deleting the app that took down slide over seems to be the solution.