  We study spherically symmetric solutions in a four-parameter
Einstein-Cartan-type class of theories. These theories include torsion, as well
as the metric, as dynamical fields, and contain only two physical excitations
(around flat spacetime): a massless spin-2 excitation and a massive spin-2 one
(of mass $ m_2 \equiv \kappa$). They offer a geometric framework (which we
propose to call "torsion bigravity") for a modification of Einstein's theory
that has the same spectrum as bimetric gravity models. We find that the
spherically symmetric solutions of torsion bigravity theories exhibit several
remarkable features: (i) they have the same number of degrees of freedom as
their analogs in ghost-free bimetric gravity theories ( i.e. one less than in
ghost-full bimetric gravity theories); (ii) in the limit of small mass for the
spin-2 field ($ \kappa \to 0$), no inverse powers of $\kappa$ arise at the
first two orders of perturbation theory (contrary to what happens in bimetric
gravity where $1/\kappa^2$ factors arise at linear order, and $1/\kappa^4$ ones
at quadratic order). We numerically construct a high-compactness
(asymptotically flat) star model in torsion bigravity and show that its
geometrical and physical properties are significantly different from those of a
general relativistic star having the same observable Keplerian mass.
