  In this article, we investigate the Riemannian and semi-Riemannian metrics on
the base space of the Boothby-Wang fibration of a closed regular non-Sasakian
$(\kappa, \mu)$-manifold. To this end, we study a natural class of deviations
of the projection map from being (semi-)Riemannian submersions. We consider
deviations that preserve the canonical bi-Legendrian structure on the given
$(\kappa, \mu)$-manifold. We present rigidity results for Riemannian and
semi-Riemannian metrics on the base space which orthogonalize the natural
bi-Lagrangian structure induced by the $(\kappa, \mu)$-structure. This approach
gives a unified framework to analyze rigidity results in both categories. More
precisely, in the Riemannian category, we obtain uniqueness of Sasakian
structure on the given $(\kappa, \mu)$-manifold which orthogonalizes the
canonical bi-Legendrian structure. In the semi-Riemannian category, we obtain
an explicit description of the finitely many para-contact structures which
orthogonalize the canonical bi-Legendrian structure.
