One arm of mesh disfigures when posing I have a rigged model with one arm that distorts and disfigures when posing while the opposite doesn't.  The disfiguring arm doesn't have any modifiers or settings (As far as I've been able to find) that are any different from the arm that poses correctly and neither do the bones.  The only thing I can figure is that maybe it has something to do with it being a duplicated + mirrored version of the side that works correctly, although the normals have all been recalculated and the mesh origins are correct.

I'm really at a complete loss here.

Here's a gif of whats happening:
Have you removed doubles before you rigged the mesh? 

Before ever rigging, you should always as a rule remove doubles in your mesh. 
Anymore when you finished modelling, select your entire mesh, press hotkey W and select "Remove Doubles." That way, you will have leas tearing of topology and less disfigurements too.