The involution of a clock diagram is the mirror image of the clock diagram, so for example major triads and minor triads are related by transposition and involution to other major and minor triads. If the involution of a set class is the same set class, it is symmetric. For example, the diatonic scale is its own involution, so is symmetric.
Tymoczko (2011) mentions that involution preserves the overall feeling of chords, but tends to change the character of melodies more, citing a melody in Bach’s Well-Tempered Clavier that is played “upside-down” the second time it is heard. Outside of twentieth century atonal music, involution is not used so much as a compositional tool as an analytical one.
See also Equivalence Groups.