Sign averaging of a sum of squares

A polynomial invariant under changing the sign of each coordinate can have its sum of squares representation averaged over independent Rademacher random variables ${\epsilon }_i\in \{-1,1\}$. Distinct parity patterns of monomials have zero cross terms because $\mathbb{E}[{\prod }_i{\epsilon }_i^{a_i}]$ vanishes when some exponent is odd. For a quadratic homogeneous polynomialthis givesThis identity separates the even monomials from each distinct two-coordinate parity pattern.