I have some understanding of harmonics but I always hear about the 1st, 2nd, etc., harmonic. Do harmonics happen an either side of the signal? Is there a -1, -2, -3, etc., harmonic or does the 1st harmonic encompass the "harmonic" on either side?

 

A harmonic is a multiple of a frequency. There are no negative frequencies.

This seems reasonable: https://en.wikipedia.org/wiki/Harmonic

 

Sometimes we refer to subharmonics, which would be 1/2, 1/3, 1/4 etc of the frequency.

 

Here is a square wave which is a sum of the fundamental frequency plus infinite odd multiples of that frequency. You can use a low pass filter to select the fundamental frequency while rejecting the harmonics (overtones) or you can use a high pass filter to ignore the fundamental while selecting any harmonic of interest. The result is a sine wave as the other harmonic (or subharmonic) components of the original wave are removed. The same method allows microcontrollers to generate a much faster clock signal than the crystal can provide (its fundamental frequency) by selecting one of the overtones.

Furthermore, the timbre of a sound is the harmonic content of a waveform that gives every sound its distinct quality. A trumpet played at 440Hz sounds different than a tuba at 440Hz because of the harmonic content.

 

or does the 1st harmonic encompass the "harmonic" on either side?

if you mean a sub-harmonic below the primary frequency, those aren't normally generated by the non-linearities that generate harmonics.