  We prove that any contact metric $(\kappa,\mu)$-space $(M,\xi,\phi,\eta,g)$
admits a canonical paracontact metric structure which is compatible with the
contact form $\eta$. We study such canonical paracontact structure, proving
that it verifies a nullity condition and induces on the underlying contact
manifold $(M,\eta)$ a sequence of compatible contact and paracontact metric
structures verifying nullity conditions. The behavior of that sequence, related
to the Boeckx invariant $I_M$ and to the bi-Legendrian structure of
$(M,\xi,\phi,\eta,g)$, is then studied. Finally we are able to define a
canonical Sasakian structure on any contact metric $(\kappa,\mu)$-space whose
Boexkx invariant satisfies $|I_M|>1$.
