I realize I'm coming a little late to the negative harmony scene, but after seeing everyone get excited it about it, I thought people would appreciate a breakdown of the basics. If you haven't watched the interviews with Jacob Collier, definitely check those out to see what sparked people's recent interest.
The idea behind negative harmony is that there is a inversion-related world of harmony that can be accessed through inverting notes around an axis. An axis is an imaginary center point, either a single note or two notes a half step apart (more on the two notes in a bit), which allows you to measure the intervals above and below it. When you invert a pitch around an axis you simply measure the interval between that note and the axis and then find the note that is equally distant from the axis in the opposite direction. For example, if your axis is the pitch C and you want to invert the pitch D, you measure the interval between C and D, which D is a whole step above C. Then you find the pitch a whole step below C, which is Bb. Likewise, a major third above C (E) would invert into a major third below C (Ab), and so on. What is above is mirrored below through inversion, and vice versa.
In negative harmony, you use an axis that allows us to keep the same tonality. In other words, when you invert the I chord, you want to it to stay a I chord, more or less. For C major, this axis is the pair of notes Eb and E. When you measure intervals above the axis you use the higher pitch, E, and when you measure intervals below you use the lower pitch, Eb. Also, when you invert E, it turns into Eb, and vice versa. This example shows all of the notes in C major (above), starting with the axis pitches, and each notes inversion (below).
Using this axis, a C major triad inverts into a C minor triad, and so a I chord turns into a i chord, keeping our general tonal center stable. This allows us to invert other chords in progressions that arrive to a I chord and still get good voice leading. More on that later. In the example below, you can see how each note inverts. The G inverts to a C, the C inverts to a G, and the E inverts to an Eb.
It's interesting and helpful to note that all major triads invert into minor triads no matter the axis. This is because the intervals simple get flipped. If we step away from negative harmony for a moment and return our axis on the note C, you'll find that the notes of a C major triad, C, E, and G invert into C, Ab, and F, the notes of an F minor triad. Where before there was a major 3rd above the axis a minor third above that, now there is a major third below the axis and a minor third below that. Similarly, under any inversion all major chords become minor chords and all minor chords become major.
Now you have the tools to invert any chord in C negative harmony, but to make things kind of easier, here is a chart showing the inversion of all the diatonic triads in C major negative harmony.
One of the important features of negative harmony, is that the voice leading for a chord progression stays the same, in a way, under inversion. In the example below I have a ii-V-I and its inversion, a bVII-vi-i. From the ii to the V chords you have one common tone and two pitches that move upward by whole step. Similarly, in negative harmony you have one common tone and two pitches that move downward by whole step from the bVII chord to the iv chord. From the V chord to the I chord there is one common tone, one pitch moves upward by half step, and one pitch moves upward by whole step. In the inversion, iv to i, there is one common tone, one pitch moves downward by half step, and one pitch moves downward by whole step. The voice leading is similar, though the direction the voices move and which members of the chord move do change.
Sixth and Seventh Chords
If we extend the harmony there are some basic patterns there too. Major sevenths invert into minor sixths and vice versa, and minor sevenths invert into major 6ths and vice versa. For example, Cmaj7 inverts into Cmin(b6), and C7 inverts into Cmin6 as shown below.
The point of all of this is to experiment and find new chord progressions. My example here is based on a ii-V-I. I inverted the ii-V and got Bbmaj6 and Fmin6 and then left my tonal center C major alone. This example only uses diatonic chords in the original progression but you can invert chromatic chords too like V7 of ii or tritone substitutions, etc. Try things out and see what you like. Enjoy!