  In this paper we study the foliated structure of a contact metric
$(\kappa,\mu)$-space. In particular, using the theory of Legendre foliations,
we give a geometric interpretation to the Boeckx's classification of contact
metric $(\kappa,\mu)$-spaces and we find necessary conditions for a contact
manifold to admit a compatible contact metric $(\kappa,\mu)$-structure. Finally
we prove that any contact metric $(\kappa,\mu)$-space $M$ whose Boeckx
invariant $I_M$ is different from $\pm 1$ admits a compatible Sasakian or
Tanaka-Webster parallel structure according to the circumstance that $|I_M|>1$
or $|I_M|<1$, respectively.
