Yep, they’re the only ones, and as mentioned earlier they’re a part of the single family that I mentioned; it’s actually quite easy to prove that.Terrific!

Doing a large simulation to get a more reliable estimate (about 18 million random perms) I get a chance of 1 in 833 for a cycle 1260. So the estimate would be 4.325e19/833 = 5.19e16. Comparing to your 5.15e16, I'd call bingo within the MOE.

So the answer is 51,490,480,088,678,400.

A question I would have is are the four basic types seen in the simulation the only ones possible? They are certainly the vast majority.

Try to get all the factors of 1260 as orientation and permutation cycles, while keeping the number of swaps even, and keeping CO and EO possible.