eleven,

You've noticed something about class 8 which I have also done, and they are definitely related to same thing.

The 16 permutations are indeed split into two "halves", elements 1 to 8 all have inverses in 1-8, and same for 9-16, which corresponds (I think) with your observation.

That makes sense, as all symmetries have transpositions with same property. I expect that every class will exhibit this property.

Nice observation!

BTW, let's put aside the question of which transforms apply to SudokuP, just for the moment, and focus on the general case.

Given a grid A which is invariant under T(8,1) = 1st transformation of class 8, how do we generate 15 grids that are invariant under T(8, 2), T(8, 3) etc

And will this method generate a distinct set for every A with given property? That would mean a complete set of invariants for class 8 can be generated from the base set of all A with T(8,1)(A) = A.