The rotational Hamiltonian of an asymmetric-top molecule, containing terms up to sixth degree in the components of the total angular momentum, is transformed by a unitary transformation to a reduced Hamiltonian, so as to avoid the indeterminacies inherent in fitting the complete Hamiltonian to observed energy levels. Two methods of reduction are considered, one suitable for energy calculations by matrix diagonalization and the other for calculations by perturbation theory. The relations between the coefficients in the reduced Hamiltonians and those in the original Hamiltonian are given. Extension of the perturbation treatment to the first-order contributions from the sextic terms and the second-order contributions from the quartic terms yields explicit expressions for these contributions in terms of Ray's function EJτ(κ) and its first and second derivatives with respect to the asymmetry parameter κ.