Thursday night Mr. Sweetie called me and told me that one of our rats, About (of the infamous Out and About) was acting funny. Her left eye was droopy, and she had no balance. He thought she might have had a stroke.
We talked to Hastur, who has more experience with rats than us, and she recommended a good vet. We weren't able to get time off on Friday to drop her off, but we called the vet that evening and she said to go ahead and drop them both off - she'd see them the same night!
So it turns out that what happened is that About got a middle ear infection somehow. She's got antibiotics to take, and in about a week should be fine.