$$1524901344$$
Calculated by C source code using exhaustive search, downloadable from my blog post. The program works in small constant space (thanks to sorear for this insight), by comparing each generated Latin square with first row fixed to its canonical representation. The canonical representation of a Latin square in this sense is the lexicographically least of the $8$ Latin squares corresponding to the $4$ rotations and $4$ reflected rotations which are then modified by permuting their symbols such that their first row becomes $(1 2 \ldots n)$. The program counts the Latin squares that are equal to their canonical representative, which thereby counts the equivalence classes as desired.