The axioms that characterize the generalized Gini social evaluation orderings for one-dimensional distributions are extended to the multidimensional attributes case. A social evaluation ordering is shown to have a two-stage aggregation representation if these axioms and a separability assumption are satisfied. In the first stage, the distributions of each attribute are aggregated using generalized Gini social evaluation functions. The functional form of the second-stage aggregator depends on the number of attributes and on which version of a comonotonic additivity axiom is used. The implications of these results for the corresponding multidimensional indices of relative and absolute inequality are also considered.