  We have previously proposed that asymptotically AdS 3D wormholes and black
holes can be analytically continued to the Euclidean signature. The analytic
continuation procedure was described for non-rotating spacetimes, for which a
plane t=0 of time symmetry exists. The resulting Euclidean manifolds turned out
to be handlebodies whose boundary is the Schottky double of the geometry of the
t=0 plane. In the present paper we generalize this analytic continuation map to
the case of rotating wormholes. The Euclidean manifolds we obtain are quotients
of the hyperbolic space by a certain quasi-Fuchsian group. The group is the
Fenchel-Nielsen deformation of the group of the non-rotating spacetime. The
angular velocity of an asymptotic region is shown to be related to the
Fenchel-Nielsen twist. This solves the problem of classification of rotating
black holes and wormholes in 2+1 dimensions: the spacetimes are parametrized by
the moduli of the boundary of the corresponding Euclidean spaces. We also
comment on the thermodynamics of the wormhole spacetimes.
