MathNet 2006 — Part 04 Question 24

Let $n$ be a positive integer divisible by $4$. Find the number of permutations $\sigma$ of $(1, 2, 3, \ldots, n)$ which satisfy the condition $\sigma(j) + \sigma^{-1}(j) = n + 1$, for all $j \in \{1, 2, 3, \ldots, n\}$.