He was called "Sundance", not the Sundance Kid and he did not wear mirrors on his hat. Apparently the mirror myth grew up because with their little black and white televisions and the show's cheap production values (mostly wide master shots) many viewers never got a close look at his hat's band. In at least one episode, reflections off the silver coins aid him in a gunfight. Not that he really needs any help as he is a faster gun, a better shot, and handier in a fist fight than anyone he ever comes up against.
The series begins with "Sundance" returning to Georgetown, Colorado after his release from prison for killing a man named Deveraux. He goes into the hotel business with the wife (Jeanette Nolan) and daughter of the man he killed; eventually becoming half owner of the Hotel de Paree, starts a stage line to Denver, and acquires a half share in a silver mine. The two women have outrageous French accents. The mother is secretly in love with Sundance and charms him with her cooking. The daughter is brazenly in love with Sundance and charms him with her more obvious assets.
Some comic relief is provided by the two parrots and by a shaggy dog named Useless. The English-speaking parrot is named Patsy and it tries to teach the French-speaking parrot English.
Then again, what do I know? I'm only a child.