  General Relativity (GR) exists in different formulations, which are
equivalent in pure gravity. Once matter is included, however, observable
predictions generically depend on the version of GR. In order to quantify the
resulting ambiguity, we employ metric-affine gravity, which encompasses as
special cases the metric, Palatini, Einstein-Cartan and Weyl formulations. We
first discuss the interaction of fermions with torsion and non-metricity, also
commenting on projective symmetry. With a view towards the Standard Model, we
then construct a generic model of (complex) scalar, fermionic and gauge fields
coupled to GR and derive an equivalent metric theory, which features numerous
new interaction terms. As a first observable consequence, we point out that a
gravitational mechanism for producing dark matter in the form of singlet
fermions can be used to distinguish between metric gravity and other
formulations of GR.
