  We give two generalizations, in arbitrary depth, of the symmetry phenomenon
used by Ball-Rivoal to prove that infinitely many values of Riemann $\zeta$
function at odd integers are irrational. These generalizations concern multiple
series of hypergeometric type, which can be written as linear forms in some
specific multiple zeta values. The proof makes use of the regularization
procedure for multiple zeta values with logarithmic divergence.
