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The modern semiclassical theory of a Bloch electron in a magnetic field now encompasses the orbital magnetic moment and the geometric phase. These two notions are encoded in the Onsager-Lifshitz quantization rule as a phase ( λ ) that is subleading in powers of the field; λ is measurable as the phase offset of magnetic quantum oscillations, e.g., in the de Haas-van Alphen effect and in tunneling spectroscopy. In some solids and for certain field orientations, λ/*π* are robustly integer-valued owing to the symmetry of the extremal orbit, i.e., they are the topological invariants of magnetotransport. Our comprehensive symmetry analysis identifies solids in any (magnetic) space group for which λ is a topological invariant; this analysis is simplified by our formulation of ten (and only ten) symmetry classes for closed, Fermi-surface orbits.

The second half of my talk focuses on spin (or pseudospin) degeneracies in the Landau-level spectrum. To tune to such degeneracies in the absence of crystalline point-group symmetries, three real parameters are needed. We have exhaustively identified all symmetry classes of cyclotron orbits for which this number is reduced from three, thus establishing symmetry-enforced 'non-crossing rules' for Landau levels. In particular, only one parameter is needed in the presence of spatial rotation or inversion; this single parameter may be the magnitude or orientation of the field.

Experimental implications are discussed for 3D Dirac-Weyl metals, crystalline and Z_{2} topological insulators, and the Rashba-Dresselhaus 2DEG subject to an arbitrarily-oriented magnetic field.