  We prove that any non-Sasakian contact metric (\kappa,\mu)-space admits a
canonical \eta-Einstein Sasakian or \eta-Einstein paraSasakian metric. An
explicit expression for the curvature tensor fields of those metrics is given
and we find the values of \kappa and \mu for which such metrics are
Sasaki-Einstein and paraSasaki-Einstein. Conversely, we prove that, under some
natural assumptions, a K-contact or K-paracontact manifold foliated by two
mutually orthogonal, totally geodesic Legendre foliations admits a contact
metric (\kappa,\mu)-structure. Furthermore, we apply the above results to the
geometry of tangent sphere bundles and we discuss some topological and
geometrical properties of (\kappa,\mu)-spaces related to the existence of
Eistein-Weyl and Lorentzian Sasakian Einstein structures.
