Retrograde inversion

In music theory, retrograde inversion is a musical term that literally means "backwards and upside down": "The inverse of the series is sounded in reverse order.

In his late twelve-tone works, however, Igor Stravinsky preferred the opposite order, so that his row charts use inverse retrograde (IR) forms for his source sets, instead of retrograde inversions (RI), although he sometimes labeled them RI in his sketches.

One work in particular by the latter composer, Nummer 2, employs retrograde of the recurring twelve-tone row B–F♯–F–E–G–A♭–E♭–D–A–B♭–D♭–C in the piano part.

[6] It is performed in both styles,[clarification needed] particularly in the outer sections of the piece.

The final movement of Paul Hindemith's Ludus Tonalis, the Postludium, is an exact retrograde inversion of the work's opening Praeludium.