If you want to go further down the rabbit hole of symmetrical scales, checkout Olivier Messiaen's modes of limited transposition https://en.wikipedia.org/wiki/Mode_of_limited_transposition. For a given set of pitches within an octave there are a limited number of times those pitches can be transposed before you wind up with the same set of pitches. And the modes in that scale must also be fewer in number than the number of pitches in the scale, meaning at least two modes of the scale must have the same interval spelling. The simplest example is the whole tone scale. Up a half step I get the same set of pitches, another half step and I get the same pitches I started with, so it is 'limited' to one transposition. And there is only one mode of the whole tone scale, since no matter where I start I always have the same set of intervals.