Particles resulting from the breaking of symmetries are arranged symmetrically about mass levels that descend in geometric progression from the Planck Mass within three sequences, of common ratio 1/π, 2/π and 1/e. Particles arranged in this way include the weak gauge bosons W^± and Z⁰, the quark weak isospin doublets u-d, s-c and b-t, and all hadronic isospin doublets. Many pairs of hadrons with some common quark content also form symmetric partnerships. Often, the mass difference characterising a partnership is precisely equal to the mass of a level. The spin-0 partners of the quarks and charged leptons are identified. The mass sequences may derive from the geometry of compact extra spaces of Planck scale.