If you ignore the issue of legality, the number of unique positions up to symmetry is relatively easy to count using

Burnside's lemma. For a combination of pieces without pawns, the group of symmetries of a chess board is D4. This group has 8 elements:

- identity;

- horizontal and vertical reflections;

- two diagonal reflections;

- rotation by 90 and 270 degrees;

- rotation by 180 degrees.

The number of unique positions up to symmetry with a chosen combination of pieces is the number of orbits of D8 acting on the set of all possible placements of those pieces on a chess board.

According to Burnside's lemma, this number is equal to the average number of placements fixed by an element of D8.

Let's consider KvK. There are 64*63 ways to place these on a chess board. The number of placements fixed by each element:

- identity: 64 * 63

- horizontal / vertical: 0

- diagonals : 8 * 7 (both Ks must be on the diagonal)

- rotation by 90/270: 0

- rotation by 180: 0

So the number of orbits is (64 * 63 + 2 * 8 * 7) / 8 = 518.

For suicide chess these are all legal. For regular chess obviously not.

To count only legal orbits, we should only look at legal placements. Unfortunately this gets nasty:

- identity: 4 * 60 + 24 * 58 + 36 * 55 = 3612

- diagonals: 2 * 6 + 6 * 5 = 42

So the number of orbits is (3612 + 2 * 42) / 8 = 462.

It should be possible to do this by hand for 3 and 4 pieces, but after that it seems to get too messy.