Let v denote the spectral radius of J A B , the block Jacobi iteration matrix. For the classes of (1) nonsingular M-matrices and (2) p-cyclic, p ⩾ 3, consistently ordered matrices, we study domains in the ( v, ω) plane when v < 1, where the block SSOR iteration method has at least as favorable an asymptotic rate of convergence as the block SOR method. Let L A ω and S A ω denote, respectively, the block SOR and SSOR iteration matrices. For the class of nonsingular M-matrices A, we determine conditions when the spectral radii satisfy ρ( S A ω) ⩽ ρ( L A ω) ∀0 < ω ⩽ 2 1 + v and ∀0 ⩽ v < 1 . Under these conditions we also show that the optimal SOR iteration parameter is ω b = 1. For the class of p-cyclic, p ⩾ 3, consistently ordered matrices A we determine for which ω's and v's, ρ( S A ω) < |ω − 1| [ ⩽ ρ( L A ω)]. Our investigations make use of the equality case in Wielandt's inequality between the spectral radii of a complex matrix and its nonnegative and irreducible majorizers and of Rouché's theorem for the location of zeros of complex functions.