[raphaelss]

It's not exactly what you are saying, but working with the complement of a given set of pitch classes (that was used in the previous section/bar or in another voice for example) is common in twelve-tone music.

So, if, for instance, an instrument is playing a melody that uses the black keys exclusively, an accompaniment (or a contrasting section) could be made that used only the white keys. The result tends to be dissonant, but it isn't necessarily dissonant to the extreme or in absolute.

Unrelated note on terminology: By inversion of a melody, we usually mean playing it upside down. So F A G would become F Db Eb. It's contour is inverted. This is done as a means of development of the melodic idea and/or in fugues/canons/"counterpointistic music in general".