This has intrigued me for awhile and I've never seen anyone else mention it.

Pipers hold the bag under their left arm, apparently uniformly so.

Most of the images I've seen of pipers, at least pre mid 19th century show the pipes being held under the right arm. This is also a common way of showing the medieval pipes being held.
Why the switch around?
Can modern pipes be held under the right arm or does the arrangement of pipes require them to be under the left? I see reenactors carry them under the left as is the modern way which seems wrong compared with the norms for the time.

I wondered if maybe holding the bag under the left was something introduced by the army to facilitate some kind of ease while marching with certain kit.

Two examples of what I mean can be seen in the Laird of Grant's Piper (1714) and Highland Wedding at Blair Atholl (1780). Bag under the right arm seems to be the norm so I wonder when it changed.