What is it called when there's 3 part harmony, and each part goes to the part above them(the highest part goes to the octave above the lowest part)? For example, in this video: at 1:16, the highest part is the melody (Hal-le-lu-jah is sung on the notes re re re do) the part below that is (ti ti ti la) and the lowest part would be (sol sol sol me). But then when the chorus comes again at 2:47, the highest part is singing sol sol sol me(but an octave higher than the lowest part before) the part below that is now the melody, and the lowest part is now singing ti ti ti la. Is there a name for this?
In general, moving the part on the bottom to the top is called inversion. The term is commonly used with intervals and chords, but it also appears in the term invertible counterpoint, where it typically refers to two-voice counterpoint in which the bottom voice may be raised by an octave (or the top voice lowered by an octave) with the resulting counterpoint still being correct according to the rules. Most notably, invertible counterpoint cannot contain parallel fourths, which are otherwise allowed, because they turn into parallel fifths on inversion.
As far as I'm aware, there's no more specific term describing the inversion you outline in the question, nor any term specific to inversions appearing in three- or four-voice part writing.
It's a common procedure termed "voice crossing." Melodic lines cross each other.
1In this case, though, it seems that there is no crossing, but rather inversion.– phoogSep 12, 2020 at 18:17