Most of the Tex-Mex styles don't use the left side for sound. Many Tex Mex boxes have the reeds removed on the left for faster bellows response and more free air.

Nice to know, Craig.
But how about the one I once saw: Santiago Jimenez?
He used his left hand (at least when he left stage and came to play in the audience.

Gus

Some do, many don't. Those who can usually play other styles as well. If you get a chance to listen to Ruben Coe, please do. He has no recordings but shows up on YouTube from time to time. One of his boxes is jazz tuned which sounds great! An excellent musician he is.
Craig

One big question left open: Why do Cajun bands rarely record the bass and chords? CDs by Horace Trahan, Ray Abshire, Steve Riley, Jimmy Breaux (and others) typically don't include this sound. Any thoughts on why it appears to be an exception to record the left hand parts? The new Les Amis Creole kickin' new CD is an exception, where you can pick out Ed Poullard's left hand pretty much throughout.