I asked a vague question about torus eversion earlier, with no hard math, so while I'm at it, how about this one, which may involve hard math:
"Everybody knows" that Stephen Smale showed us how to evert a sphere without tearing or creasing:
http://www.youtube.com/watch?v=R_w4HYXuo9M
Has anyone done the same with a torus?
This seems to suggest the answer is yes. Séquin goes in to some detail in this document and I'll add the image he references, made by Cheritat.