How would I identify what's causing my Mac to permanently hang on shutdown?
My first response whenever something is broken in Mac OS X is to try a safe boot. Safe boots perform several repair tasks in addition to safe booting your mac without extensions or startup items running.
Shut down your Mac. Power it up and begin holding down the shift key the moment you hear the startup chime. Keep it down until you see an onscreen progress bar. Eventually, you will get a login screen - Login and wait until your machine is fully started up. Now restart normally and see if problem is fixed.