A novel singular gauge transformation is developed for quasiparticles in the mixed state of a strongly type-II superconductor which permits a full solution of the problem at low and intermediate fields. For a periodic vortex lattice the natural quasiparticle states are the Bloch waves and {\em not} Landau levels discussed in the recent literature. The new representation elucidates the physics considerably and provides a number of fresh insights into the spectral and transport properties of such systems.