Number of permutations of [2n] where $x_i + x_{i+1} \ne 2n+1$

Question

As stated in title, what is the number of permutations of $[2n]$ where $x_i + x_{i+1} \ne 2n+1 \;,\forall i\in[2n-1]$. I want to use the inclusion-exclusion theorem, and consider separately permutations with $x_i + x_{i+1} \ne 2n+1$ for a certain $i$, but the intersection of these depend on the indices and seem hard to compute.