co.combinatorics – Is there a short (conceptual) way to prove that this combinatorial map is invariant by cyclic permutation?

Consider a positive integer $n$ and integers $c_i$, $i=1,2,3,4$, such that $1 le c_i le n$. For all $c_4$, consider the map:

$$m_{c_4}: (c_1,c_2,c_3) mapsto delta_{c_1,c_2}delta_{c_3,c_4} – # { |2n+1-2|x||, x in {c_1+c_2, c_3+c_4, c_1-c_2, c_3-c_4} },$$

where the notation $#$ means cardinal. The problem is to show that this map is invariant by cyclic permutation, i.e.
$$ m_{c_4}(c_1,c_2,c_3) = m_{c_4}(c_2,c_3,c_1). $$

I see a straightforward way to prove that by cases, but it should be quite long.

Question: Is there a short (conceptual) way to prove that?

Context: this problem poped up to prove that some rings (parametrized by $q$) are associative.

Just for the conveniance of the reader, here is the checking for $n<20$:

sage: def map(c1,c2,c3,c4,n):
....:     return kronecker_delta(c1,c2)*kronecker_delta(c3,c4)-len(set((abs(2*n+1-2*abs(x)) for x in (c1+c2,c3+c4,c1-c2,c3-c4))))
sage: for n in range(1,20):
....:     for c1 in range(1,n+1):
....:         for c2 in range(1,n+1):
....:             for c3 in range(1,n+1):
....:                 for c4 in range(1,n+1):
....:                     if map(c1,c2,c3,c4,n)!=map(c2,c3,c1,c4,n):
....:                         print((c1,c2,c3,c4,n))