  The paper is a complete study of paracontact metric manifolds for which the
Reeb vector field of the underlying contact structure satisfies a nullity
condition (the condition \eqref{paranullity} below, for some real numbers $%
\tilde\kappa$ and $\tilde\mu$). This class of pseudo-Riemannian manifolds,
which includes para-Sasakian manifolds, was recently defined in \cite{MOTE}. In
this paper we show in fact that there is a kind of duality between those
manifolds and contact metric $(\kappa,\mu)$-spaces. In particular, we prove
that, under some natural assumption, any such paracontact metric manifold
admits a compatible contact metric $(\kappa,\mu)$-structure (eventually
Sasakian). Moreover, we prove that the nullity condition is invariant under $%
\mathcal{D}$-homothetic deformations and determines the whole curvature tensor
field completely. Finally non-trivial examples in any dimension are presented
and the many differences with the contact metric case, due to the non-positive
definiteness of the metric, are discussed.
