Let $J$ be a permutation of the first $N$ integers (1, 2, …, $N$), so that the permuted sequence reads $(J(1),J(2),…,J(N))$. The function $J$ must of course be a bijection. Additionally, suppose that $J$ is its own inverse: $J(J(i))=i$, and that $J$ has, at most, one fixed point. That is, there is either none or one value of $i$ such that $J(i)=i$. It isn’t hard to see that there is a single fixed point if and only if $N$ is odd, and there is none if $N$ is even.
A permutation $J$ with the above properties establishes a pairing of the integers from 1 to $N$, where $i$ is paired with $J(i)$ (except if $N$ is odd, in which case the fixed point is not paired).
Is there a name for this type of permutation?
I would call this a fixed-point free involution, respectively involution with one fixed point.
After some searching I found the term “maximum matching of the complete graph on $N$ points” that unambiguously describes the set of edges that bijectively corresponds to a permutation of this type (you could also say “maximal” instead of “maximum”, as this is the same thing for a complete graph). This avoids the even/odd dichotomy, but has the drawback of only somewhat indirectly describing a set of permutations.