  In a recent paper [1], it was introduced a new class of gravitational
theories with two local degrees of freedom. The existence of these theories
apparently challenges the distinctive role of general relativity as the unique
non-linear theory of massless spin-2 particles. Here we perform a comprehensive
analysis of these theories with the aim of (i) understanding whether or not
these are actually equivalent to general relativity, and (ii) finding the root
of the variance in case these are not. We have found that a broad set of
seemingly different theories actually pass all the possible tests of
equivalence to general relativity (in vacuum) that we were able to devise,
including the analysis of scattering amplitudes using on-shell techniques.
These results are complemented with the observation that the only examples
which are manifestly not equivalent to general relativity either do not contain
gravitons in their spectrum, or are not guaranteed to include only two local
degrees of freedom once radiative corrections are taken into account. Coupling
to matter is also considered: we show that coupling these theories to matter in
a consistent way is not as straightforward as one could expect. Minimal
coupling, as well as the most straightforward non-minimal couplings, cannot be
used. Therefore, before being able to address any issues in the presence of
matter, it would be necessary to find a consistent (and in any case rather
peculiar) coupling scheme.
