The number of symmetrically distinct solutions to the $n$-queens problem is Sequence number A002562 in the On-Line Encyclopedia of Integer Sequences. The sequence from $n=1$ to $n=26$ is 1, 0, 0, 1, 2, 1, 6, 12, 46, 92, 341, 1787, 9233, 45752, 285053, 1846955, 11977939, 83263591, 621012754, 4878666808, 39333324973, 336376244042, 3029242658210, 28439272956934, 275986683743434, 2789712466510289

The number of solutions to the $n$-queens problem is Sequence number A000170 in the On-Line Encyclopedia of Integer Sequences. The sequence from $n=1$ to $n=26$ is 1, 0, 0, 2, 10, 4, 40, 92, 352, 724, 2680, 14200, 73712, 365596, 2279184, 14772512, 95815104, 666090624, 4968057848, 39029188884, 314666222712, 2691008701644, 24233937684440, 227514171973736, 2207893435808352, 2231769961636404.