Progress in shader land.
Okay, maybe not. I didn't realize it at the time, but a whole bunch of normal calculations were reversed (or rather, the algorithms I was trying to use expected things pointing in the other direction). This is partially why the interior of the fabric geometry is visible from the outside, which clearly shouldn't be possible if the surface of the fabric was actually opaque (which it should have been).
Nevertheless, this lead to a really bizarre satin-like effect that a friend would later describe as "classy". While it was technically nothing more than a happy accident, at least the weave pattern stuff seemed to be working at this point.