Beds and Desks (noch nicht übersetzt)

At Euler University, each of the $n$ students (numbered from 1 to $n$) occupies a bed in the dormitory and uses a desk in the classroom.

Some of the beds are in private rooms which a student occupies alone, while the others are in double rooms occupied by two students as roommates. Similarly, each desk is either a single desk for the sole use of one student, or a twin desk at which two students sit together as desk partners.

We represent the bed and desk sharing arrangements each by a list of pairs of student numbers. For example, with $n=4$, if $(2,3)$ represents the bed pairing and $(1,3)(2,4)$ the desk pairing, then students 2 and 3 are roommates while 1 and 4 have single rooms, and students 1 and 3 are desk partners, as are students 2 and 4.

The new chancellor of the university decides to change the organisation of beds and desks: a permutation $\sigma$ of the numbers $1,2,\ldots,n$ will be chosen, and each student $k$ will be given both the bed and the desk formerly occupied by student number $\sigma(k)$.

The students agree to this change, under the conditions that:

1. Any two students currently sharing a room will still be roommates.
2. Any two students currently sharing a desk will still be desk partners.

In the example above, there are only two ways to satisfy these conditions: either take no action ($\sigma$ is the identity permutation), or reverse the order of the students.

With $n=6$, for the bed pairing $(1,2)(3,4)(5,6)$ and the desk pairing $(3,6)(4,5)$, there are 8 permutations which satisfy the conditions. One example is the mapping $(1, 2, 3, 4, 5, 6) \mapsto (1, 2, 5, 6, 3, 4)$.

With $n=36$, if we have bed pairing:
$(2,13)(4,30)(5,27)(6,16)(10,18)(12,35)(14,19)(15,20)(17,26)(21,32)(22,33)(24,34)(25,28)$
and desk pairing
$(1,35)(2,22)(3,36)(4,28)(5,25)(7,18)(9,23)(13,19)(14,33)(15,34)(20,24)(26,29)(27,30)$
then among the $36!$ possible permutations (including the identity permutation), 663552 of them satisfy the conditions stipulated by the students.

The downloadable text files beds.txt and desks.txt contain pairings for $n=500$. Each pairing is written on its own line, with the student numbers of the two roommates (or desk partners) separated with a comma. For example, the desk pairing in the $n=4$ example above would be represented in this file format as:

1,3
2,4

With these pairings, find the number of permutations that satisfy the students' conditions. Give your answer modulo $999\,999\,937$.