I put the snows on my 2015 Rogue a few days ago. The tires are Hakkapeliitta 8 studded snows on dedicated steel wheels. The tires have a directional tread. My summer tires are on their own wheels too.

After install I noticed that the rotation reminders are all pointing in the wrong direction. I've changed them myself since 2015, except for in 2020 when I had major heart surgery and had to bring them into the shop to be swapped out.

I've always marked the tires with a tire crayon at each switch over (DF, DR, PF, PR) to make sure that they stayed on the proper side and only got rotated front to rear.

So, they've either been installed on the wrong side of the vehicle since new (which I doubt), or since 2020 (2 full winters of use) when the shop changed them.

My questions and concerns are .... since they've been rolling backwards for at least 2 winters and possibly more, is it a bad idea to swap them to the proper side and get them rotating as per the rotation arrows? I was under the impression that you can do damage to steel belted radials if you switch their direction of rotation (supposedly this didn't apply to bias belted radials).

Thoughts? Any science to this?

Belts taking a "set" is less of an issue than it used to be, but you're still taking a risk if you switch them back. If it does create an issue, the tire will generally give you plenty of warning. You'll get a deformity that first makes it noisy and then devolves into a noticeable out-of-round condition, usually accompanied by a thump-thump-thump as the belt degenerates. Tires mounted with the rotation backwards are generally just inefficient at squeezing out water, so you might just want to leave well enough alone and be wary in the rain.

Agreed, directional tread is generally for water evacuation, to push water out the sides.
Very heavy rain and wet roads will increase hydroplane as water will feed the center of the tire.

Thats what I figured in regards to evacuating water. The sipes in the tread which are basically in a V shape, are forcing water towards the center of the tread as opposed to pushing it outwards to the sides.

I cannot afford to start replacing tires right now as if one goes bad I'll probably need to replace all four to get the outside diameter the same on all four. I've read that different diameters between new and used, even slight, can play havoc with AWD systems.