  We study the Cox ring of the moduli space of stable pointed rational curves,
\M_{0,n}, via the closely related permutohedral (or Losev-Manin) spaces. Our
main result establishes \binom{n}{2} polynomial subrings of the Cox ring, thus
giving collections of boundary variables that intersect the ideal of relations
trivially. As applications, we give a combinatorial way to partially solve the
Riemann-Roch problem for \M_{0,n}, and we show that all relations in degrees of
Cox(\M_{0,6}) arising from certain pull-backs from projective spaces are
generated by the Plucker relations.
