Almost all Chrysler products of the 1956-1964 era had HVAC buttons on the right-hand side of the steering wheel that essentially mirrored the shift-control buttons on the left-hand side of the wheel. This produced a fairly balanced, symmetrical look to the dashboard.
In many cases, the slide control for HVAC temperature (on the right) even mirrored the lever for the “Park” position on the transmission (on the left). In fact, the '63 Plymouth had this exact type of symmetry. The photos at the link provided by Anno perfectly refreshed my memory my Dad’s '63 Belvedere–on which I learned to drive!
However, it was possible to switch the positions!
If presented with medical evidence of a physical handicap involving one’s left hand, Chrysler would reverse the above-noted positions, thus putting the trans controls on the right. (I have no idea whether there was a charge for this change, but I suspect that there was a fee involved)
If–and only if–Anno’s brother had one of these handicapped-adapted cars would the shift-control buttons have been on the right side of the steering wheel.