So I think I'm using a single atlas page (if by atlas page you mean one image that has all the parts of the sprite). And restarting Unity definitely doesn't fix the problem.
What does eventually fix it is deleting all of the assets and recreating them. Sometimes I have to go through multiple passes. It's possible I'm misconfiguring things, but it seems like Unity is just having difficulty in general with atlas assets.
Additionally, and this is strange, sometimes it reads the atlas correctly (i.e. doesn't reproduce the entire sprite sheet somewhat grotesquely for every limb), but certain slot images will be upside down.
So I guess the summary of the issue is, inconsistent behavior when initially creating the atlas asset. Since it only seems to change after deleting everything and re-pasting it in, and I think it's specifically in the atlas asset, maybe it's an issue in the creation, not the configuration, of the Atlas .asset file?
I guess after puzzling through all of this I might have enough information to have an idea of where to look in the source code to figure the issue out, but if not, I hope this issue report has been helpful...
