  We show that the phase space of three-dimensional gravity contains two layers
of dualities: between diffeomorphisms and a notion of "dual diffeomorphisms" on
the one hand, and between first order curvature and torsion on the other hand.
This is most elegantly revealed and understood when studying the most general
Lorentz-invariant first order theory in connection and triad variables,
described by the so-called Mielke-Baekler Lagrangian. By analyzing the
quasi-local symmetries of this theory in the covariant phase space formalism,
we show that in each sector of the torsion/curvature duality there exists a
well-defined notion of dual diffeomorphism, which furthermore follows uniquely
from the Sugawara construction. Together with the usual diffeomorphisms, these
duals form at finite distance, without any boundary conditions, and for any
sign of the cosmological constant, a centreless double Virasoro algebra which
in the flat case reduces to the BMS$_3$ algebra. These algebras can then be
centrally-extended via the twisted Sugawara construction. This shows that the
celebrated results about asymptotic symmetry algebras are actually generic
features of three-dimensional gravity at any finite distance. They are however
only revealed when working in first order connection and triad variables, and a
priori inaccessible from Chern-Simons theory. As a bonus, we study the second
order equations of motion of the Mielke-Baekler model, as well as the on-shell
Lagrangian. This reveals the duality between Riemannian metric and teleparallel
gravity, and a new candidate theory for three-dimensional massive gravity which
we call teleparallel topologically massive gravity.
