We show that degree-d block-symmetric polynomials in n variables modulo any odd p correlate with parity exponentially better than degree-d symmetric polynomials, if n ≥ cd 2 log d and d ∈ [0.995·p t −1, p t ) for some t ≥ 1 and some c > 0 that depends only on p. The result is obtained through the development of a theory we call spectral analysis of symmetric correlation, which originated in works of Cai, Green, and Thierauf [CGT96,Gre99]. In particular, our result follows from a detailed analysis of the correlation of symmetric polynomials, which is determined up to an exponentially small relative error when d = p t − 1. *