  On-shell constructibility is redefining our understanding of perturbative
quantum field theory. The tree-level S-matrix of constructible theories is
completely determined by a set of recurrence relations and a reduced number of
scattering amplitudes. In this paper, we revisit the on-shell constructibility
of gravitational theories making use of new results on soft theorems and
recurrence relations. We show that using a double complex shift and an all-line
soft deformation allows us to relax the technical conditions for
constructibility, in order to include more general propagators and
higher-derivative interactions that prevent using conventional
Britto-Cachazo-Feng-Witten (BCFW) shifts. From this result we extract a set of
criteria that guarantee that a given gravitational action has the same
tree-level S-matrix in Minkowski spacetime as general relativity, which implies
the equivalence at all orders in perturbation theory between these classical
field theories on asymptotically flat spacetimes. As a corollary we deduce that
the scattering amplitudes of general relativity and unimodular gravity are the
same for an arbitrary number of external particles (as long as the S-matrix of
the latter is unitary), thus extending previous works that were able to deal
only with $n=4$ and $n=5$ amplitudes.
