  Tri-bimaximal mixing is a specific lepton mixing ansatz, which has been shown
to account very successfully for the established neutrino oscillation data.
Working in a particular basis (the `circulant basis'), we identify three
independent symmetries of tri-bimaximal mixing, which we exploit to set the
tri-bimaximal hypothesis in context, alongside some simple, phenomenologically
interesting CP-conserving and CP-violating generalisations.
