  We present a classification of the complete, simply connected, contact metric
$(\kappa,\mu)$-spaces as homogeneous contact metric manifolds, by studying the
base space of their canonical fibration. According to the value of the Boeckx
invariant, it turns out that the base is a complexification or a
para-complexification of a sphere or of a hyperbolic space. In particular, we
obtain a new homogeneous representation of the contact metric
$(\kappa,\mu)$-spaces with Boeckx invariant less than $-1$.
