The pistol was worn on the right hip , butt forward so the Sabre could be used with the right hand and the pistol (revolver) in the left. This doctrine continued , according to the Ordnance Manuals, into the 1900s.
The Army (or Navy or Marines) didn't care if you were right or left handed .
It kinda feels like the manuals are worded like you weren't supposed to really aim the revolver it was for point shooting at close range and the Sabre was the primary close quarters weapon.
I don't think revolvers even received any official mention in an Ordnance Manual until the 1860s, the single shot pistol was still the "norm" in the doctrine through the 1850s with revolvers kinda being a special item I guess.
This is why the cap cutout is on the right side of the gun.