In Copenhagen Metro on every escalator you can see 2 labels:
I had an impression that prepostion "to" is used when you describe a direction of the moving action. In that sense, how can you stand "to" something? Wouldn't it more correct to say "stand on the right side" and "walk on the left side"?
It is correct (by common usage?).
Seems like it is an easy way of saying standers move to the right side and walkers move to the left side.
bonus read: http://standtotheright.com/
To the right is a phrase on its own and means on or towards the right.
In this sense you can easily say stand to the right.
Yes, 'to' can be used in such cases.
"Stand to the right" as you say means "stand on the right," or another way of putting it, "stand towards the right." "Towards" in this case doesn't mean moving in a direction, just more on one side, or "stand more on the right side."
Other cases where 'to' is used are "Stand to attention." It doesn't mean direction of moving, just directing the person what to do.
"Stand to the right" is a perfectly normal phrase - because the action involves staying or being in relation to something else. I could, for example stand to the right of you.
"Walk to the left" is rather more suspect. Although it is grammatically correct - I could walk to the left of a room - what that would mean is that I would be travelling towards the region or object labelled "the left". To indicate that my path of travel is on the left of something, I would say that I "walk on the left".
London uses "stand on the right" on its signs, but the proposed Transport for London byelaw 9.1 says "Persons shall keep to the right of escalators when not walking up, down or along them".
Before the introduction of comb escalators, the sign at the end said "Step off: right foot first", apparently difficult for those used to military training.
