The impact of the symmetry of the reaction-coordinate dependence of the solvent friction on the thermally activated barrier crossing rate is examined. Possible symmetry forms are defined for the reaction-coordinate dependence of the solvent friction. The implications in the effective Grote-Hynes theory of Voth [J. Chem. Phys. 97, 5908 (1992)] and the theory recently presented by Haynes, Voth, and Pollak [J, Chem, Phys. 101, 7811 (1994)] of a spatially antisymmetric solvent friction are illustrated. Surprisingly, no correction to the Kramers-Grote-Hynes theory for the transmission coefficient is predicted, although an antisymmetric spatial dependence of the solvent friction is a strong departure from the usual spatially independent friction-based generalized Langevin equation, The results from the analytical theories are compared to numerically exact generalized Langevin equation simulation results for a simple model system and found to agree well for a wide range of damping strengths and friction time scales, confirming the interesting predictions of these theories.