  The main goal of this paper is a calculation of the integral (co)homology of
the group of symmetric automorphisms of a free product. We proceed by giving a
geometric interpretation of symmetric automorphisms via a moduli space of
certain diagrams, which we name cactus products. To describe this moduli space
a theory of diagonal complexes is introduced. This offers a generalisation of
the theory of right-angled Artin groups in that each diagonal complex defines
what we call a diagonal right-angled Artin group (DRAAG).
