Is there a name for regular bands that satisfy $xyx=yx$ for all $x$,$y$?
A right regular band. The dual notion of left regular bands has been used to study random walks on hyperplane arrangements and oriented matroids. The paper of Ken Brown on Semigroups, rings and Markov chains is a good place to start. My paper with Margolis and Saliola gives more details on the representation theory as does Saliola's solo paper. 

