  The asymptotic structure of the Pauli-Fierz theory at spatial infinity is
investigated in four spacetime dimensions. Boundary conditions on the massless
spin-$2$ field that are invariant under an infinite-dimensional group of
non-trivial ``improper'' gauge symmetries are given. The compatibility of these
boundary conditions with invariance of the theory under Lorentz boosts is a
subtle issue which is investigated in depth and leads to the identification of
the improper gauge symmetries with the pure BMS supertranslations. It is also
shown how rigid Poincar\'e transformations and improper gauge symmetries of the
free Pauli-Fierz theory merge into the full BMS group as one switches on the
gravitational coupling. Contrary to the massless spin-$1$ case, where
invariance under boosts is implemented differently and where important
differences between the free and the interacting cases have been exhibited
recently, the free Pauli-Fierz theory and general relativity show very similar
behaviours at spatial infinity.
