It is a classical fact that the cotangent bundle $T^* M$ of a differentiable manifold $M$ enjoys a canonical symplectic form $\Omega^*$. If $(M,j,g,\omega)$ is a pseudo-Kähler or para-Kähler $2n$-dimensional manifold, we prove that the tangent bundle $T M$ also enjoys a natural pseudo-Kähler or para-Kähler structure $(J,G,\Omega)$, where $\Omega$ is the pull-back by $g$ of $\Omega^*$ and $G$ is a pseudo-Riemannian metric with neutral signature $(2n,2n)$. We investigate the curvature properties of the pair $(J,G)$ and prove that: $G$ 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 check that (i) the holomorphic sectional curvature of $(J,G)$ is not constant unless $g$ is flat, and (ii) in $n=1$ case, that $G$ is never anti-self-dual, unless conformally flat.