  We construct a scattering theory for the spin $\pm2$ Teukolsky equations on
the exterior of the Schwarzschild spacetime, as a first step towards developing
a scattering theory for the linearised Einstein equations in double null gauge.
This is done by exploiting a physical-space version of the Chandrasekhar
transformation used by Dafermos, Holzegel and Rodnianski in [arxiv:1601.06467]
to prove the linear stability of the Schwarzschild solution. We also address
the Teukolsky-Starobinsky correspondence and construct an isomorphism between
scattering data for the $+2$ and $-2$ Teukolsky equations. This will allow us
to state an additional mixed scattering statement for a pair of curvature
components satisfying the spin $+2$ and $-2$ Teukolsky equations and connected
via the Teukolsky-Starobinsky identities, completely determining the radiating
degrees of freedom of solutions to the linearised Einstein equations.
