The basis of a polychord voicing is to play two different chords at the same time, such as one in the left hand and one in the right on a piano. The relationship between the two chords determines the quality of the resultant chord. These are always two handed voicings on a piano, or five or six string voicings on the guitar. They produce a very rich, complex sound compared to the voicings presented so far.
The simplest style of polychord voicing is to play two triads; for instance, a C major triad in the left hand on a piano, and a D major triad in the right. This will be notated D/C. This notation is overloaded in that it is usually interpreted as meaning a D triad over the single note C in the bass; it is not always clear when a polychord is intended. Polychords are seldom explicitly called for in written music, so there is no standard way to notate them. You must normally find your own opportunities to play polychords.
If you take all the notes in this D/C voicing and lay them in a row, you will see that this describes either the C lydian or C lydian dominant scales. Therefore, this voicing can be used over any chord for which those scales are appropriate. If you experiment with other triads over a C major triad, you will find several combinations that sound good and describe well known scales. However, many of these combinations involve doubled notes, which can be avoided as described below. Among the polychords that do not involve doubled notes are Gb/C, which produces a C HW diminished scale, Bb/C, which produces a C mixolydian scale, Dm/C, which produces a C major or C mixolydian scale, Ebm/C, which produces a C HW diminished scale, F#m/C, which also produces a C HW diminished scale, and Bm/C, which produces a C lydian scale. These polychords may be used as voicings for any chords that fit the corresponding scales.
You may have noticed that Db/C, Abm/C, Bbm/C, and B/C also involve no doubled notes and sound very interesting, although they do not obviously describe any standard scales. There are no rules for when these polychords may be played as voicings. When your ear becomes accustomed to the particular nuances and dissonances of each, you may find situations in which you can use them. For example, the last polychord listed, B/C, sounds good when used as a substitute for Cmaj7, particularly in the context of a ii-V-I progression, and especially at the end of a song. You may resolve it to a normal Cmaj7 voicing if you wish.
You can construct similar polychords with a minor triad at the bottom. Db/Cm produces a C phrygian scale; F/Cm produces a C dorian scale; Fm/Cm produces a C minor scale; A/Cm produces a C HW diminished scale; Bb/Cm produces a C dorian scale; and Bbm/Cm produces a C phrygian scale. In addition, D/Cm produces an interesting, bluesy sounding scale.
I mentioned before the desire to avoid doubled notes. One way to construct polychords that avoid doubled notes is to replace the triad at the bottom with either the third and seventh, the root and seventh, or the root and third of a dominant chord. Voicings constructed in this fashion are also called upper structures. They always imply some sort of dominant chord.
For example, there are several possible C7 upper structures. A Dbm triad over “C Bb” yields a C7b9#5 chord. A D triad over “E Bb” yields a C7#11 chord. An Eb triad over “C E” yields a C7#9 chord. An F# triad over “C E” yields a C7b9b5 chord. An F#m triad over “E Bb” yields a C7b9b5 chord. An Ab triad over “E Bb” yields a C7#9#5 chord. An A triad over “C Bb” yields a C7b9 chord.
You will find it takes a lot of practice to become familiar enough with these voicings to be able to play them on demand. You may wish to choose a few tunes and plan ahead of time where you will use these voicings. It is well worth the effort. The richness and variety introduced by these voicings can add a lot to your harmonic vocabulary.