Since the 3rd and 7th of a dominant seventh chord are a tritone apart, and a tritone is the exact bisect of an octave, the inversion of these two pitches will form the 3rd and 7th of another dominant seventh chord. Another way of looking at this is by measuring from the root of the chord. Dominant seventh chords who's roots are a tritone apart will share the same pitches for their 3rd and 7th. The rootless shell of one chord will be the inverted rootless shell of the other chord. In other words, the 3rd of one dominant seventh chord will be the 7th of the other dominant seventh chord in this pairing, and the 7th will be the 3rd of the other chord.
This unique relationships causes the two chords to be interchangeable harmonically. The tritone partner chord of any V7 can be used as a substitute for dominant harmony within that key. The use of this type of harmonic substitution is called tritone substitution. This technique is particularly useful in ii-V-I chord progressions. It allows for more variety within the progression, and a bass line that moves chromatically instead of having cycle of 5th motion. The included video demonstrates the use of tritone substitution in both major and minor ii-V-I progressions.
This Learning Music With Ray video discusses tritone substitution in dominant harmony. It is a study of the unique relationship between dominant seventh chords who's roots are a tritone apart. In this video, I discuss the fact that the tritone is the exact midpoint of an octave. This causes the interval's inversion to also be a tritone. Then I discuss the impact that this symmetry has on dominant harmony. Finally, I provide examples of tritone substation in ii-V-I chord progressions on the piano keyboard.