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Composite fermions (CFs) of the fractional quantum Hall effect are describedas spherical products of electron and vortex spinors, built from underlyingL=1/2 ladder operators aligned so that the spinor angular momenta Le and Lv aremaximal. We identify the CF's quantum numbers as the angular momentum L in (L_eL_v)L, its magnetic projection m_L, the electron number N, with L_v={N-1)/2,and magnetic \nu-spin, m_\nu=L_e-L_v. Translationally invariant FQHE states areformed by filling p subshells with their respective CFs, in order of ascendingL for fixed L_e and L_v, beginning with the lowest allowed value, L=|m_\nu|. Weshow that this wave function has an exactly equivalent hierarchical form.  FQHE states can be grouped into \nu-spin multiplets mirror symmetric aroundm_\nu=0, with N held constant. Electron particle-hole conjugation with respectto this vacuum is identified as the mirror symmetry relating FQHE states of thesame N but distinct fillings \nu = p/(2p+1} and p/( 2p-1).  Alternatively, mirror symmetric \nu-spin multiplets can be constructed inwhich the magnetic field strength is held fixed: the valence states areelectron particle-vortex hole excitations. Particle-hole symmetry -- relatingthe N-particle FQHE state of filling \nu=p/(2p+1} to the $\bar{N}$-particlestate of filling {p+1)/(2p+1} -- is shown to be equivalent to electron-vortexexchange.  In this construction $\bar{N}$-N CFs of the higher density state occupy anextra zero-mode subshell. We link this structure, familiar from supersymmetricquantum mechanics, to the CF Pauli Hamiltonian, which we show is isospectral,quadratic in the \nu-spin raising and lowering operators, and four-folddegenerate. On linearization, it takes a Dirac form similar to that found inthe integer quantum Hall effect (IQHE).

Particle-Hole Mirror Symmetries around the Half-Filled Shell: The Quantum Numbers and Algebraic Structure of Composite Fermions