50/60Hz Noise Harmonics
I am communicating through air (wireless) using pure sinusoids via a
  magnetic transmitter and receiver (i.e. near-field)


Chances are that the interference you receive is 60 Hz AND odd harmonics. This statement is based on the fact that the current taken by localized circuits (such as transformers) will be somewhat rich in odd harmonics due to core saturation.

Most common powerful loads/components such as transformers and induction motors tend to take a symmetrical (same shape for positive and negative half cycles) but distorted current. When the waveform is symmetrical there tends to be only odd harmonics. Taken to extremes a square wave only has odd harmonics: -




  Why are the harmonics of the signal I am transmitting not visible
  (e.g. 3rd harmonic of 125Hz signal is 375Hz)?


Because you are transmitting pure sinusoids and they don't have harmonics. Of course if you don't change frequency seemlessly then you'll see some harmonics creeping in.