Involutions in which all transpositions have the same distance

Question

I am wondering whether there exist a name in the literature for the following special type of permutation $\pi$ over $\{1, \dots n\}$. $\pi$ is an involution, i.e., it is composed of nonoverlapping transpositions $(a_i, b_i)$. Now, also the distance between the end-points in each transposition should be the same, i.e., $|a_i - b_i| = k$ for all $i$ for some $k$.