The idea is to find assignments to the first three rows of the Sudoku that yield the same number of completed sudokus.
For every assignment of the first three rows, we assign a canonical form by first applying the unique permutation on the digits $1,\ldots,9$ such that the first square is $1,2,3;4,5,6;7,8,9$, then rearranging the other 6 columns in some canonical way (see details in the paper). Canonicalization doesn't change the number of completed sudokus.
Other operations that don't change the number of completed sudokus is permutations of the rows or the columns. The idea is to start with a list of all canonicalized first three rows, and identify items which are related by row/column permutation. You do this by going over all items, applying all possible permutations, and canonicalizing; now you can identify the item you started with with the item you ended with.