  Let $M$ be an $n$-dimensional manifold with a torsion free affine connection
$\nabla$ and let $T^*M$ be the cotangent bundle. In this paper, we consider
some of the geometrical aspect of a twisted Riemannian extension which provide
a link between the affine geometry of $(M,\nabla)$ and the neutral signature
pseudo-Riemannian geometry of $T^*M$. We investigate the spectral geometry of
the Szab\'o operator on $M$ and on $T^*M$.
