Also:

1 2 3
4 5 6
8 9 10

1 2 3
5 6 7
9 10 11

1 2 4
5 6 8
9 10 12
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Any additive offset to a row or column from a valid state that doesn't cause a value collision also satisfies the permutation condition.

3 6 9
2 5 8
1 4 7

1 2 3
7 8 9
4 5 6

2 1 3
5 4 6
8 7 9
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Rotations, row swaps and column swaps also yield valid results. Of course, these are actually a subset of the additive transformation. If you think about it, the 1 .. 9 square is just the all 1's square subjected to 9 row additions and 3 column additions; the minimum necessary number to achieve element uniqueness.

The real question for me is are there valid results which are not mappable via addition to the base square.