As familiar as I was with the circle of fifths, I just realized the obvious fact that you can obtain it from the "circle of semitones" ¹ just by:
* Splitting it into the circle of "even notes" (C D E F# Ab Bb) and of "odd notes" (C# Eb F G A B)
* Rotating the circle of odd notes
* Putting the two circles back together
(I don't know that my description is clear, maybe it needs a visualization too 😅 the idea is simple enough, though)
¹ That is, the circle of pitch classes where each note's neighbors are the notes a semitone away from it)