There is a well-known paper by Russell & Jarvis that describes a method of arriving at the conclusion that "There are 5472730538 essentially different Sudoku grids".
They define a "Sudoku Symmetry Group" as a permutation group $G$, ie a subgroup of $S_{81}$, with elements corresponding to the allowable grid re-arrangements, such as permuting bands, stacks, sets of 3 rows/columns, etc.
Two grids are regarded as essentially different if one can't be transformed into the other by the defined permutations (+ relabelling).
In a nutshell they used Burnside's Lemma to arrive at the result, by counting the automorphic Sudoku grids only for a single representative of each conjugacy class of $G$.
There is a "Restricted Sudoku" variant of this puzzle in which, as well as the normal rules (different values in every row, column and 3x3 block) we add a 4th constraint that the values in all cells that are in the same position within each of the blocks should all be different.
This effectively adds a 4th region to which a cell belongs.
For example, this grid has the given property. The set of cells corresponding to position (1,1) within each block (marked *) are all different, and so are the sets corresponding to positions (1,2), (1,3), (2, 1) etc:
*3 9 6 | *1 7 4 | *5 8 2
1 7 4 | 5 2 8 | 3 6 9
5 2 8 | 3 9 6 | 1 4 7
------------------------
*9 6 3 | *7 4 1 | *2 5 8
4 1 7 | 8 5 2 | 6 9 3
8 5 2 | 6 3 9 | 4 7 1
------------------------
*6 3 9 | *4 1 7 | *8 2 5
7 4 1 | 2 8 5 | 9 3 6
2 8 5 | 9 6 3 | 7 1 4
We's like to know how many "essentially different" Restricted Sudokus there are, just as the authors of [1] did. We can define a simpler group $G'$ in which row/col permutations are restricted to all 3 bands or stacks.
Any grid that has the positional-difference property will retain that property when permuted under these conditions. Clearly $G'$ is also a subgroup of $S_{81}$, and can be shown to have order $2 \times 6^4$ = 2592, and is generated by just 5 transformations. (The $G$ for the normal Sudoku case has order $2 \times 6^8$ = 3359232)
A colleague suggested that there is a problem with this group, namely that Burnside's Lemma was not applicable. His concern was this: there are some grid transformations (eg: permuting just the 3 rows in a band) that do not always preserve the positional difference property.
My feeling is that this does not apply here, because isomorphism under that type of transformation is defined only for Sudoku grids, we exclude those permutations from our symmetry group. Perhaps $G'$ is not a subgroup of $G$, but it surely remains a subgroup of $S_{81}$.
So we can use Burnsides' Lemma just as the authors did for Sudoku. At least, that's the theory!
Are we on safe ground here?