Hi, all!

Just thinking about 2 ways of symmetry concepts - the first one is "human", based on geometric concepts, and second one is based on Group Theory concepts. It would be nice to find common approach ...

Let's consider solution grids only. It simplifies symmetry analysis.

First, I should say, that half turn (pi rotation) symmetry always come with double diagonal symmetry (and with vertical+horizontal symmetry, and with quarter turn symmetry).

Double diagonal symmetry (DDS) is represented by automorphisms group of order 4 (it contains 4 VPTs). Quarter turn symmetry (QTS) is represented by automorphisms group of order 4 too. What symmetry is "more symmetric"?

DDS automorphisms group contains 3 non-trivial subgroups (having 2 VPTs each), representing 3 "basic" simmetries - diagonal, antidiagonal and half turn symmetries. All its VPTs participate one of 3 subgroups. So, the group itself is not primitive, but "composed" of other basic symmetries. Totally, DDS automorphisms group contains 3 "basic" symmetries.

QTS automorphisms group contains only 1 non-trivial subgroup - half turn symmetry. 2 (out of 4) VPTs don't participate this subgroup, so we should count automorphisms group itself as "basic" symmetry. Totally, QTS automorphisms group contains 2 "basic" symmetries.

Therefore double diagonal symmetry is more symmetric than quarter turn symmetry.

Serg