  We introduce a spinorial version of the scattering equations, the
\emph{polarized scattering equations}, that incorporates spinor polarization
data. They lead to new formulae for tree-level scattering amplitudes in six
dimensions that directly extend to maximal supersymmetry. They give a quite
distinct framework from that of Cachazo et al.; in particular, the formulae do
not change character from even to odd numbers of particles. We find new
ingredients for integrands for maximally supersymmetric Yang-Mills, gravity, M5
and D5 branes. We explain how the polarized scattering equations and
supersymmetry representations arise from an ambitwistor-string with target
given by a super-twistor description of the geometry of super-ambitwistor space
for six dimensions. On reduction to four dimensions the polarized scattering
equations give rise to massive analogues of the 4d refined scattering equations
for amplitudes on the Coulomb branch. At zero mass this framework naturally
generalizes the twistorial version of the ambitwistor string in four
dimensions.
