We show how all known N 2, d 4, 5, 6 maximally supersymmetric vacua (Hpp-waves and aDS × S solutions) are related through dimensional reduction/oxidation preserving all the unbroken supersymmetries. In particular, we show how the N 2, d 5 family of vacua (which are the near-horizon geometry of supersymmetric rotating black holes) interpolates between aDS₂ × S³ and aDS₃ × S² in parameter space and how it can be dimensionally reduced to an N 2, d 4 dyonic Robinson–Bertotti solution with geometry aDS₂ × S² and oxidized to an N 2, d 6 solution with aDS₃ × S³ geometry (which is the near-horizon limit of the self-dual string).