The E and C collections can be swapped by inversions around E, F, A or C (the tones common to both scales). Similarly, the C and D collections can be swapped by inversions around E, G, B/A, D/C and the D and E collections by inversions around D, F, A, or B. All other transformations do not change the classes (e.g. reflecting the E collection around E gives the E collection once again). This unfortunately means that the inversions do not act as a simple cyclic group on the set of diminished scales.
The E and C collections can be swapped by inv!ersions around E, F, A or C (the tones common to both scales). Similarly, the C and D collections can be swapped by inversions around E, G, B/A, D/C and the D and E collections by inversions around D, F, A, or B. All other transformations do not change the classes (e.g. reflecting the E collection around E gives the E collection once again). This unfortunately means that the inversions do not act as a simple cyclic group on the set of diminished scales.