How is it that the magnetism from the coil in a relay switch does not distort the signal passing through it?

 

The coil and the contacts are somewhat separated, and the signals through the contacts do not have any relationship to the energizing current in the coil.

Passing a signal in the vicinity of an magnetic field has no effect on the signal, as the magnetic field is concentrated by the pole piece.

http://en.wikipedia.org/wiki/Relay

 

Relays operating with a DC coil don't affect the signal because the magnetic field does not vary with time. The only time it can create noise in the signal is when it switches on/off.
Relays operating with AC can add noise to the signal conducted by the relay. However, relays are designed in such a way that the magnetic field is weak in the vicinity of the relay conducts. Also, the length of the conducts is short and thus the induced voltage will be tiny.