We determine the Kähler and quaternionic manifolds of the untwisted multiplets of superstrings compactified on Z N ⊂SU(3) symmetric orbifolds. In particular we show that, when moduli and family replicas are considered, there is a nontrivial interference between the massless modes related (1, 1) and (2, 1) harmonic forms: the manifolds naturally depend on the net chiral family content and on the family-antifamily paired multiplets.