The ambiguities of the effective Hamiltonians for isolated doubly excited vibrational states of bending modes of tetrahedral molecules are investigated theoretically. Reduced forms of these Hamiltonians and invariant parameters are proposed. The theoretical conclusions are tested by actual fits of experimental data for the 2v2,vIn2 +v4and 2v4bands of12CH4. The experimental behaviours of fitted parameters are found to be in satisfactory agreement with theory. The results obtained, which represent one step of the comprehensive study of the so-called pentad, lead to significant improvements not only in the convergence of least squares but also in the accurate understanding of the physical meaning of specroscopic parameters. A generalization to states includingnquanta of thev1mode is outlined.