  Theories of gravity invariant under those diffeomorphisms generated by
transverse vectors, $\pd_\m\xi^\m=0$ are considered. Such theories are dubbed
transverse, and differ from General Relativity in that the determinant of the
metric, $g$, is a transverse scalar. We comment on diverse ways in which these
models can be constrained using a variety of observations. Generically, an
additional scalar degree of freedom mediates the interaction, so the usual
constraints on scalar-tensor theories have to be imposed. If the purely
gravitational part is Einstein--Hilbert but the matter action is transverse,
the models predict that the three {\em a priori} different concepts of mass
(gravitational active and gravitational passive as well as inertial) are not
equivalent anymore. These transverse deviations from General Relativity are
therefore tightly constrained, actually correlated with existing bounds on
violations of the equivalence principle, local violations of Newton's third law
and/or violation of Local Position Invariance.
