The turbolift set was composed of 6 curved and 3 flat panels, plus a set of sliding doors which would be simply butted up again an existing set of doors on the Bridge or corridor.
In their standard configuration they looked like this:
However, when assembled for use on set this arrangement was not always followed. There are plenty of instances where the motion indicator panel ended up on the rear centre:
https://tos.trekcore.com/hd/albums/1x04hd/thenakedtimehd0617.jpg
https://tos.trekcore.com/hd/albums/3x13hd/elaanoftroyiushd0038.jpg
https://tos.trekcore.com/hd/albums/3x07hd/dayofthedovehd0279.jpg
Sometimes a partial construction seems to have been used, probably to allow the actors to walk on and off set. However the flats were not always aligned correctly to the doors, as seen in this particularly egregious example from Season One and a similar one from Season Three:
I could be wrong but I think the only time controls were seen on left hand panel (when looking into the turbolift) was in
Wolf In The Fold, probably because the plot directly called for them. Most likely the right hand panel was just swapped to the left. I say this because when that side of the set was filmed a few episodes later, no controls were to be seen!
In-universe however I am more than happy to imagine the existence of different styles of turbolift cruising the shafts around the ship. It would also help to explain the numerous doorway inconsistencies (which I'm not going to go on about right now, don't worry!)