  We study Lagrangians with the minimal amount of gauge symmetry required to
propagate spin-two particles without ghosts or tachyons. In general, these
Lagrangians also have a scalar mode in their spectrum. We find that, in two
cases, the symmetry can be enhanced to a larger group: the whole group of
diffeomorphisms or a enhancement involving a Weyl symmetry. We consider the
non-linear completions of these theories. The intuitive completions yield the
usual scalar-tensor theories except for the pure spin-two cases, which
correspond to two inequivalent Lagrangians giving rise to Einstein's equations.
A more constructive self-consistent approach yields a background dependent
Lagrangian.
