  Starting from $g$-natural pseudo-Riemannian metrics of suitable signature on
the unit tangent sphere bundle $T_1 M$ of a Riemannian manifold
$(M,\langle,\rangle)$, we construct a family of paracontact metric structures.
We prove that this class of paracontact metric structures is invariant under
$\mathcal D$-homothetic deformations, and classify paraSasakian and paracontact
$(\kappa,\mu)$-spaces inside this class. We also present a way to build
paracontact $(\kappa,\mu)$-spaces from corresponding contact metric structures
on $T_1 M$.
