The paper investigates the assignment of items to groups such that all groups are as heterogeneous as possible and each pair of groups is as balanced as possible, i.e. all groups get the same number of items assigned, the total diversity is maximized, and all pairs of groups are as homogeneous as possible regarding the attribute values of their assigned items. We present a two-step exact mixed-integer programming approach, whereat the first step is a maximally diverse grouping problem (MDGP), i.e. our approach extends the MDGP by a balancing step. Moreover, we present a detailed complexity analysis for both steps and heuristic solution approaches based on 2-opt, very large-scale neighborhood search, and variable neighborhood search, which are, together with the mixed-integer program, evaluated in a computational study.