Nobody has really studied this area as far as I know.

For standard Sudoku (9 x 9), it is highly likely that the determinants of ED grids are unique, but that's about all one can say.

[

EDIT ] Wrong (again) ... see post below

This fairly recent

paper looks interesting, if only you could get access to it without paying for the privilege.

[

EDIT ] Now available, see post below

A quote from the abstract:

We show that non-trisotopic latin squares with equal permanents and equal determinants exist for all orders n≥9 that are divisible by 3.

A new word for us - 2 Latin Squares are

trisotopic if they are isotopic, or transpose-isotopic. This corresponds to our Sudoku definition of

essentially different.