  We study the geometry of non-minimal surfaces of supercritical constant mean
curvature invariant under screw motions in the homogeneous 3-manifolds
$\mathbb{E}(\kappa,\tau)$ including the space-forms of non-negative curvature.
We give a complete classification, thereby unifying and extending various
previous results. We give the first classification for the Berger sphere case,
and we exhibit a new family of screw motion CMC surfaces, called tubes.
