We present magnetization measurements on the single molecule magnet Mn6, revealing various tunnel transitions inconsistent with a giant-spin description. We propose a dimeric model of the molecule with two coupled spins S=6, which involves crystal-field anisotropy, symmetric Heisenberg exchange interaction, and antisymmetric Dzyaloshinskii-Moriya exchange interaction. We show that this simplified model of the molecule explains the experimentally observed tunnel transitions and that the antisymmetric exchange interaction between the spins gives rise to tunneling processes between spin states belonging to different spin multiplets.