We study the strongly correlated two-dimensional Hubbard model for general band filling by a transformation of the model to a spin-free representation. A pair transformation is constructed that maps a strong-coupling Hubbard problem in which the particles obey Fermi statistics onto a spin-free representation in which the charge degrees of freedom are spinless fermions, while constraining the physical system to an eigenstate of total spin. The Gutzwiller constraint is thus satisfied kinematically in the transformed problem for a class of Pauli-allowed many-fermion spatial wave functions. We find that the required pair transformation within the spin-singlet sector leads to correlated many-body states with broken time-reversal symmetry. For small band filling in the spin-singlet sector, the transformed problem for the charge degrees of freedom is formally identical to a problem for indistinguishable noninteracting 1/2 anyons (semions). For small deviations from half filling, the charge degrees of freedom (holes) in the singlet sector are again naturally represented as 1/2 anyons but in the presence of a strong uniform background U(1) gauge field. Exactly at half filling the charge degrees of freedom are quenched, and the formulation reduces to the spin-singlet wave function for the two-dimensional lattice first proposed by Kalmeyer and Laughlin. The one- and two-hole dynamics around this neutral chiral state on the two-dimensional lattice are considered in detail. Finally, numerical calculations are reported to test the accuracy of these states as approximations to the ground state of the Hubbard Hamiltonian. Over a range of band fillings we find that Gutzwiller projections of free-particle determinants provide improved hole kinetic energies, while a projection of the flux phase provides improved spin-exchange energies for the pure Hubbard problem.