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 bilegendrian structure on the given \((\kappa ,\mu )\) -manifold. This approach gives a unified framework to analyze rigidity results in both categories. As a consequence, in the Riemannian category, we obtain uniqueness of Sasakian structure on the given \((\kappa ,\mu )\) -manifold which orthogonalizes the canonical bilegendrian structure. In the semi-Riemannian category, we obtain an explicit description of the finitely many para-Sasakian structures which orthogonalize the canonical bilegendrian structure.
