I hope this question has some sort of meaningfulness. Suppose you are on the phone with an alien which is on his planet. For some reason he know which are our UP and DOWN and our FRONT and BACK. It's not difficult to explain him where is the UP or where is the BACK in a "physical way".
Today I found out that there is a way to explain the alien the LEFT and RIGHT too, by means of some decaying process, but now I'm interested in a more mathematical way to do that.
I suppose that it can't be done because there is an "unbreakable symmetry" between the two direction.
How to prove the impossibility of solving the problem? How could the problem be modeled in a more abstract one?
I make this reasoning: i tell to the alien to think of the floor as a plane and to draw a line on it which have the direction of his nose; then, maintaining his own orientation draw another line perpendicular to the first and chose an arbitrary direction for it. Let's call this direction A and the opposite would be B. Now, if a can make "operations" on A and B that converge to my RIGHT (but there has not to be reference to my left/right in these operations) I would have done. But this can't be done, I think.
For operations I mean, for instance, to pick a vector in the plane, rotating it, etc..
Whether this is possible depends on the shape of the universe, in the following sense.
Suppose the alien has guessed some answer. How could you confirm with the alien that it was correct? Well, the alien could travel to where you are, and you could compare. But it turns out that depending on the shape of the universe, whether you and the alien agree can depend on what path it takes to get to you.
More precisely, the issue is that the universe might fail to be orientable. This is easiest to visualize on a Möbius strip. If you and an alien are on opposite sides of a Möbius strip, then whatever you both think left, right, up, and down mean, whether your answers agree or not depends on whether the alien approaches you from one direction or the other. (Draw a picture! Or watch this lovely video by Vi Hart.)