Is this a mutual inductor? If not, what is it and how does it work. 
I am not sure what a "mutual inductor" is.
To me, it looks like a relay with a regular coil inside. For a relay, I doubt mutual inductance matters much since it is usually only a signal coil inside…
I agree with James.
Those inductors look to me like they are there to slow current flow when the switch is flipped.
DAB
trehill01 Nope. Mutual inductance in the influence of one inductor on another or another conductor in the circuit. In some cases (transformer, magnetic amplifier, etc), it is intentional. In other cases…