Fiber-preserving and orientation-reversing involutions of Seifert fibered 3-manifolds

We consider fiber-preserving, orientation-reversing involutions on orientable Seifert fibered 3-manifolds and the conditions on a manifold for admissibility of such involutions. We construct a class $Ψ$ of fiber-preserving, orientation-reversing involutions that act trivially on the base. Each element of $Ψ$ is obtained by extending a product involution across Seifert pieces of type $V(2,2;-1)$ - a solid torus with three fibers filled according to Seifert invariants $(2,1)$, $(2,1)$, and $(1,-1)$. We show that $Ψ$ forms a single conjugacy class under fiber-preserving diffeomorphisms. Our main result establishes that any fiber-preserving, orientation-reversing involution factors as $ψ\circ g$, where $g$ is fiber-preserving and orientation-preserving and $ψ\inΨ$, thus reducing the problem to the previously known orientation-preserving case. Through the orientable base-space double covering, we further extend the classification to manifolds with non-orientable base orbifold.