It would be easy to speculate, but who knows the real reason, if there was one? Chuck Connors was left-handed, so maybe his natural tendency was to shoot left-handed and he would forget. Maybe he was tasked to shoot right- or left-handed depending on the camera angle during a scene. His character, Lucas McCain, was an officer in an Indiana infantry regiment during the Civil War. It would behoove most any soldier to be able to shoot with either hand when necessary. Maybe nobody was noticing which hand he shot with during filming, and it was an inside joke with him. Since he obviously had a reputation as a good-guy gunman on the show, maybe that was part of the reason for the notoriety: he was ambidextrous and good with either hand. You've seen Westerns where the gunman is shooting two revolvers at the same time from a double rig. Same deal.
I don't remember a lot of the episodes, I was very young (9 when the show was cancelled in 1963), and I haven't had the opportunity to closely watch re-runs when they're shown. I don't recall seeing him carry the rifle in a scabbard when on horseback, it seems like he was usually carrying it in his hands and walking, or in a wagon. Several promotional pictures I've seen show him holding it sort of port-arms, left-handed. I've never thought much about it, really.
I remember there was a toy rifle offered when I was a kid, it had the large cocking lever, and had a small tab on the lever that could be folded out and would catch the trigger when the action was worked, so you could rapid fire it just by working the cocking lever.
The Rifleman was a good show, always had a moral, and Connors' character as a single parent was groundbreaking. As for the gunplay, I still liked Steve McQueen and his Mare's Leg in Wanted : Dead or Alive better. More 'cool factor'.
There are several "quirks" about the rifle itself. Since the show was set in 1880 it exists fully 12 years before John Browning designed it (it was a Winchester Model 1892). The magazine could not hold 12 rounds, which is how many times the gun is fired in opening credits (and there is a 13th shot dubbed as the opening music starts) The blanks used in the show had smokeless powder, where the .44-40 was a black powder cartridge and the rapid firing would have filled the scene with smoke.