6x6 Checker Challenge

Examine the $6x6$ checkerboard below and note that the six checkers are arranged on the board so that one and only one is placed in each row and each column, and there is never more than one in any diagonal. (Diagonals run from southeast to northwest and southwest to northeast and include all diagonals, not just the major two.)

Column
1 2 3 4 5 6
----------------
1 | | O | | | | |
----------------
2 | | | | O | | |
----------------
3 | | | | | | O |
----------------
4 | O | | | | | |
----------------
5 | | | O | | | |
----------------
6 | | | | | O | |
----------------

The solution shown above is described by the sequence $246135$, which gives the column positions of the checkers for each row from $1$ to $6$.

ROW $123456$
COLUMN $246135$

This is one solution to the $6x6$ Checker Challenge. Write a program that searches and finds all unique solution sequences to the $6x6$ Checker Challenge. What is the total number of solutions found (including reflections and rotations.)?

Source: 1993 USACO Qualifying Round 1