K - X Makes the Difference

You are given an integer $n$. We call a positive integer $x$ *good* with respect to $n$, if there exists a permutation$^\dagger$ $p$ of the integers $\{1, 2, \ldots, n\}$ such that $$ |p_i - p_{n + 1 - i}| = x $$ for *all* $1 \le i \le n$. Here, $|d|$ denotes the absolute value of $d$. For example, $|1| = 1, |-2| = 2, |0| = 0$. Count the number of integers that are *good* with respect to the g