  We investigate the Lagrangian formulation of the double-copy correspondence
between gauge theories and gravity, up to the cubic order. Building on the
definition of the double-copy field as a convolution of two vectors, we obtain
free gravitational Lagrangians as products of two Yang-Mills Lagrangians, in a
form amenable to be easily extended to the massive case. We discuss the origin
of these results from tensionless strings and show the existence of gauge
fixings that mix the two spin-one sectors and lead to an alternative,
especially simple, version of the free Lagrangian. We then construct cubic
vertices for the full double-copy multiplet, comprising a graviton, a two-form
and a scalar particle, by means of the Noether procedure. Both at the free and
at the cubic level the result gets uniquely fixed only upon imposing, on top of
gauge invariance, a left-right Lorentz symmetry ruling contraction of indices
among double-copy fields. Whereas the outcome nicely matches the cubic
interactions of $\mathcal{N}=0$ supergravity, including the gauge-invariant
coupling between the scalar particle and the two-form, such a twofold Lorentz
symmetry seems to conflict with the perturbative reconstruction of space-time
geometry.
