Let $(M,J,g,\omega)$ be a pseudo-Kähler or para-Kähler $2n$-dimensional manifold. We prove that the tangent bundle $TM$ enjoys a natural pseudo-Kähler or para-Kähler structure $(\tilde{J},\tilde{g},\Omega)$, where $\Omega$ is the pull-back by $g$ of the canonical symplectic structure on $T^*M$, and $\tilde{g}$ is a pseudo-Riemannian metric with neutral signature $(2n,2n)$. We investigate the curvature properties of the pair $(\tilde{J},\tilde{g})$: we prove that $\tilde{g}$ is scalar-flat, is not Einstein unless $g$ is flat, has nonpositive (resp.\ nonnegative) Ricci curvature if and only if $g$ has nonpositive (resp.\ nonnegative) Ricci curvature as well, and is locally conformally flat if and only if $n=1$ and $g$ has constant curvature, or $n>2$ and $g$ is flat. We also prove that (i) the holomorphic sectional curvature of $(\tilde{J},\tilde{g})$ is not constant unless $g$ is flat, and (ii) in $n=1$ case, that $\tilde{g}$ is never anti-self-dual, unless conformally flat.