Callan permutations and odd order permutations

We call a permutation to be of odd order if writing in cycle form consisting of only odd cycles, and call a permutation to be a Callan permutation if all its left-to-right minima appear at odd positions. This paper aims to provide five elementary proofs that Callan permutations and odd order permutations have the same cardinality: one by generating functions, two by recursions and another two by combinatorial bijections. The last bijection gives rise to a refinement of this equality.