I have played piano for about two years or so, and one of my epiphanies was when I began learning chord inversions after natural position chords were muscle memory. With the help of my scale book and some exploration, I found that if you went around the circle clockwise, you could get from C major to C major in order of playing Natural Position C major, then 1st inversion of the 5th above it (G), then 2nd inversion 5th above that (D), and landing onto a natural position chord (A). If this would make it so if you followed this formula, you would start and end exactly where we were sonically but simply moving in a Natural-1st-2nd-Natural inversion pattern.
Eventually I did this so often I could then go counter-clockwise around the circle. C major Natural Position <- F major 2nd Inversion <- Bb major 1st Inversion <- Eb major natural position.
The most important thing about this to me is it makes the circle of 5th makes sonical sense. By that I mean it sounds like playing a scale, but it's not a scale technically.
If you played these simple chords in the order I describe in a musical way with emphasis on the 1st beat of the three, it sounds magical.
C major Natural Position G Major 1st inversion D Major 2nd inversion A Major Natural Position E Major 1st inversion B Major 2nd inversion F# Major Natural Position C# Major 1st inversion Ab Major 2nd inversion Eb Major Natural Position Bb Major 1st inversion F Major 2nd inversion C Major Natural Position