The 648 comes from 18 (row perms) x 18 (col perms) x 2 (transpo). It's easier to see for a slightly reordered version of the grid, but it basically comes down to: any choice of band order (6), then any choice of cycling within all bands the same way (3), giving 18 row perms; same for columns; then similar with the grid transposed. To be sure, I actually checked the whole group of 3359232 ops and did indeed find 648 symmetries.coloin wrote:648 = ? 18*6*6 ....[could you explain these?]

I guess your method is considering small deviations from a crop of 33s, so perhaps it's unsurprising that it's rare for two slightly-different grids to be isomorphic. None of the new (isomorphic) ones got by use of symmetry share the 15-clue backbone, for example.