  We calculate the cubic order terms in a covariant theory that gives a
nonlinear completion of the Fierz-Pauli massive spin-2 action. The resulting
terms have specially tuned coefficients guarantying the absence of a ghost at
this order in the decoupling limit. We show in this limit that: (1) The
quadratic theory propagates helicity-2, 1, and helicity-0 states of massive
spin-2. (2) The cubic terms with six derivatives -- which would give ghosts on
local backgrounds -- cancel out automatically. (3) There is a four-derivative
cubic term for the helicity-0 field, that has been known to be ghost-free on
any local background. (4) There are four-derivative cubic terms that mix two
helicity-0 fields with one helicity-2, or two helicity-1 fields with one
helicity-0; none of them give ghosts on local backgrounds. (5) In the absence
of external sources, all the cubic mixing terms can be removed by nonlinear
redefinitions of the helicity-2 and helicity-1 fields. Notably, the helicity-2
redefinition generates the quartic Galileon term. These findings hint to an
underlying nonlinearly realized symmetry, that should be responsible for what
appears as the accidental cancellation of the ghost.
