Why are the periods of these permutations often 1560? I ran across a math puzzle that went like this:
Consider the list $1,9,9,3, \cdots$ where the next entry is equal to the sum mod 10 of the prior 4.  So the list begins $1,9,9,3,2,3,7,\cdots$. Will the sequence $7,3,6,7$ ever occur?
(Feel free to pause here and solve this problem for your own amusement if you desire.  Spoiler below.)
So the answer is "yes", and we can solve this by noticing that the function to derive the next digit is invertible so we can derive digits going to the left as well.  Going left, we find $7,3,6,7$ pretty quickly.
I wrote a program and found that the period (equivalently the length of the permutation's cycle) is 1560.  But surprisingly (to me) altering the starting sequence from 1,9,9,3 to most any other sequence left the period at 1560.  There are a few cases where it changes; for example, starting with 4,4,4,4 we get a period of length only 312.
So, my question: what's special about 1560 here?
Note: This feels a lot like LFSRs, but I don't know much about them.

Your recurrence is linear in that you can add two series together term by term and still have it a series.  The period of (0,0,0,1) is 1560, so all periods will be a divisor of that.  To get 1560 you just have to avoid shorter cycles.
