A topological manifold is known as orientable when it has a concept of left- and right-handedness, so that a right-handed object remains right handed as it moves around inside the space. A non-orientable space, in contrast does not have this feature. One may think of higher-dimensional analogues of the Möbius strip http://upload.wikimedia.org/wikipedia/commons/e/e7/MobiusStrip-01.png
or the Klein bottle, which are non-orientable:
http://upload.wikimedia.org/wikipedia/commons/5/5c/Klein_bottle.svg
There may be little reason at first to believe that the actual physical universe is orientable, although we do experience this as a local phenomenon. But it is conceivable that a right-handed space traveler, having followed a certain path, would return as left-handed, while still insisting that he or she is right-handed. In such a universe, there would seem to be no fact of the matter about left and right.
Meanwhile, it turns out that some experiments show that some of the fundamental particles exhibit an asymmetry in their handedness, so that the right-handed forms interact with each other in a way that is different from what one would predict from the left-handed interactions. The wikipedia page on symmetry explains that this phenomenon arises only with the weak interaction.
With respect to these theories, then, one can tell left from right by carefully observing the weak interaction of various sub-atomic particles.