The linear and nonlinear rheological behavior of dilute ferrofluids is determined from an underlying kinetic model and the dependence of the viscosity coefficient on the scalar orientational order parameters is obtained. In case of uniaxial symmetry, the antisymmetric contribution to the hydrodynamic stress tensor is of the same form as in the classical Ericksen-Leslie theory of uniaxial nematic liquid crystals and the linear magnetoviscosity is found to coincide with earlier results obtained by the so-called effective field method. While the assumption of uniaxial symmetry is fulfilled exactly in the limit of strong vorticity and weak magnetic field, the exact result for the linear magnetoviscosity shows corrections due to contributions from biaxial symmetry. Measures for the deviations from uniaxial symmetry are introduced and the generalization of the stress tensor in case of biaxial symmetry is obtained. The investigations are accompanied by numerical simulation of the kinetic equation and reveal that the assumption of uniaxial symmetry seems to be a good approximation for most values of the magnetic field and vorticity.