Update
I figure out my problem. I made an entirely new Rails app with just one model, installed the Paperclip gem and attached photos to the model. It all worked.
When I started to hunt for differences between the new app and my existing app, I noticed that I had Paperclip installed both as a gem and as a plugin in my existing app (and just as a gem in the new app).
I deleted the plugin from my existing app (leaving just the gem) and it all works now. All aspects other than deleting worked with both installed, but apparently deleting crossed the wires.