The chromatic pitch class set consists of an interesting structure. A pentatonic set is both complement and subset of the diatonic set. The compelementary pentatonic with the two tones outside both pentatonics make up another diatonic, the mirror key. So each diatonic key has a mirror key shifted by +/- 6 halftones.
Eg. Gb is mirror with C.
12-tone chromatic set:
C Db D Eb E F Gb G Ab A Bb B
C diatonic set:
C D E F G A B
Two pentatonics:
C D E G A
Gb Ab Bb Db Eb
Gb diatonic set (mirror of the C diatonic set):
Gb Ab Bb B Db Eb F
For a given key there's just a single key with the least number of shared tones (and in some sense is the most distant) - it is just its mirror key. The relation is symmetric, so there are six pairs of mirror keys in the 12-tone set.
Having the mirror key tool in your toolbox allows to make sense of many phenomena that otherwise do not make much sense in the traditional harmony.
As an example it seems that tritone substitution of the dominant chord, ie. using IIb7 in place of V7 is just a visit to the mirror key.
Other example is the Tristan chord.