tinfoil's most recent two posts are indeed along the lines of those I had been thinking. But I'm not sure whether the asserted example works or not. (One has to be more careful here than I had been in previous attempts, as I discovered to my cost!) But there are similar things which do work. Consider
123 457 689
456 189 237
789 236 145
and exchange all the 1s and 4s:
423 157 689
156 489 237
789 236 415
Any way to fill in blocks 4-9 to make a complete grid will also be a way to fill in
123 457 689
456 189 237
789 236 415
in which the 14/41 configurations in blocks 1 and 2 are exchanged. We conclude that the number of ways of completing the first and third of these grids (which differ only in that the 1 and 4 in block 3 are interchanged) are the same. (In the example, you can do something similar with the 6 and 9.)
It looks like my catalogue for blocks 4 and 7 will also be shortened a little with the aid of this trick.
I've failed to program this correctly so far. (My program recognises where this sort of thing may occur, but then doesn't do what I want it to! Maybe I'll have time tomorrow or Saturday to fix it...)
Frazer