  We study a large family of metric-affine theories with a projective symmetry,
including non-minimally coupled matter fields which respect this invariance.
The symmetry is straightforwardly realised by imposing that the connection only
enters through the symmetric part of the Ricci tensor, even in the matter
sector. We leave the connection completely free (including torsion) and obtain
its general solution as the Levi-Civita connection of an auxiliary metric,
showing that the torsion only appears as a projective mode. This result
justifies the widely used condition of setting vanishing torsion in these
theories as a simple gauge choice. We apply our results to some particular
cases considered in the literature like the so-called
Eddington-inspired-Born-Infeld theories among others. We finally discuss the
possibility of imposing a gauge fixing where the connection is metric
compatible and comment on the genuine character of the non-metricity in
theories where the two metrics are not conformally related.
