A couple weeks ago I drove my girlfriend to see her podiatrist. I liked the guy, and while I was there I asked about getting an appointment for myself. Today I had my own appointment and he identified the problem as bone spurs on my heels at the posterior end of the longitudinal arch. These develop when the ligaments and tendons start to tear away from the bones under stress. When my feet aren't under stress things start to heal, only to be torn again when I stand up and walk. Thus the ache.
The "good" news is that he doesn't blame ballet. He points his finger at the boots I wore this past winter, saying they flex in the wrong place and twist where they should be stiff. I'm about to buy a pair of new hiking boots (the soles of my old ones are badly worn) and now I know what to look for and what to avoid.
He also says that they've learned that surgery is not indicated for this sort of problem. He thinks they'll heal over time, given the opportunity. He hand-made a pair of soft arch supports from tape and padding that slide over my feet to support the arch. Although he said I could go to class wearing these supports, I'm going to be away from classes for a few weeks more for other reasons. Maybe the break will aid the healing process.