  Recently two of the authors presented a spinorial extension of the scattering
equations, the `polarized scattering equations' that incorporates spinor
polarization data. These led to new worldsheet amplitude formulae for a variety
of gauge, gravity and brane theories in six dimensions that naturally
incorporate fermions and directly extend to maximal supersymmetry. This paper
provides a number of improvements to the original formulae, together with
extended details of the construction, examples and full proofs of some of the
formulae by BCFW recursion and factorization. We show how our formulae reduce
to corresponding formulae for maximally supersymmetric gauge, gravity and brane
theories in five and four dimensions. In four dimensions our framework
naturally gives the twistorial version of the 4d ambitwistor string, giving new
insights into the nature of the refined and polarized scattering equations they
give rise to, and on the relations between its measure and the CHY measure. Our
formulae exhibit a natural double-copy structure being built from
`half-integrands'. We give further discussion of the matrix of theories and
formulae to which our half-integrands give rise, including controversial
formulae for amplitudes involving Gerbes.
