Why is it not possible to explain to an alien "at the phone" which side is left and which is the right side by defining a simple experimental setup using induction? Defining for instance downwards should not be a problem, or is it?
Say you want to explain to your Martian friend what "clockwise" means. You instruct him to look down and place a coil of wire on the floor, and connect it to the battery so that the current flows in what he hopes is clockwise (the hope being that if something comes out wrong, he can then reverse this). You can then, and you do, tell him that the magnetic field inside the coil is now pointing down. So far so good: electric current is a vector and it $+/-$ charges can be independently verified, and we can assume he knows what "down" is.
However, once he's done that, he phones back and says "OK, but how do I actually confirm that the magnetic field is pointing down?". "Well," you say, "fling a positive charge horizontally through the field. It should curve left."
... and you can imagine his response: "so what was 'left', again?".
What this illustrates is that while some electromagnetic quantities, like the magnetic field, do depend on a convention for left/right handedness, and therefore transform to themselves after a space inversion, these are not directly physically observable. All pseudo-vector quantities in electromagnetism couple through a cross product to make physical observables, which means that mirror versions of the same apparatus will have magnetic fields going opposite ways but all forces, positions and velocities will behave identically.
Of course, this is absolutely necessary. If you begin with a parity-invariant theory, then all subsequent physical observables will be parity invariants. To transmit chiral information, you need to use physical observables from a chiral theory; the standard example is the weak interaction. There you can make an apparatus that will transmit a sense of handedness to an alien friend, with the caveat that, since the weak interaction is $CP$ invariant, if your friend is made of antimatter then he'll get it the wrong way round.
I'll just finish by saying that if you didn't read about this first directly from the Feynman lecture, you really should go read that.