Summary A multivariate Hausdorff operator H = H(µ, c, A) is defined in terms of a s-finite Borel measure µ on Rn, a Borel measurable function c on Rn, and an n × n matrix A whose entries are Borel measurable functions on rn and such that A is nonsingular µ-a.e. The operator H*:= H (µ, c | det A-1|, A-1) is the adjoint to H in a well-defined sense. Our goal is to prove sufficient conditions for the boundedness of these operators on the real Hardy space H1(Rn) and BMO (Rn). Our main tool is proving commuting relations among H, H*, and the Riesz transforms Rj. We also prove commuting relations among H, H*, and the Fourier transform.