For intermolecular perturbation theories in which it is assumed that the unperturbed wave function of the composite system is a product of the unperturbed wave functions of its components, and which satisfy one general constraint, we derive two renormalized interaction energy expressions which are more accurate than the perturbation expansions, when all are evaluated to comparable order. This is accomplished by focusing on the parameter A in terms of which the perturbation expansions are derived rather than on the potential of interaction between components. In the derivation of each renormalized energy formula, we discard zeroth-through infinite-order terms which do not contribute to the interaction energy when the interaction is turned on fully, i.e., when A = 1. The first renormalized interaction energy when A = 1 is identical in form to the interaction energy in the symmetrized Rayleigh-Schrodinger (SRS) theory, but not in interpretation. The wave function appearing in the renormalized energy cannot generally be that assumed in the SRS theory, and the renormalized energy to zeroth order in A is not zero. The latter is not surprising because we discarded a zeroth-order term in the derivation. The second renormalized interaction energy formula is derived from the first by using the same set of assumptions and arguments that were used in deriving the first. We expect it to be more accurate than the first, which is expected to be more accurate than the sum of the perturbation energies, all evaluated to comparable order. These expectations are supported by the results of calculations on LiH using two perturbation theories, the polarization approximation and the Amos-Musher theory. The first-order wave functions for both were calculated in the configuration interaction (CI) approximation; then the interaction energies were calculated by summing the perturbation energies through third order and by evaluating the renormalized energy expressions. The perturbation results are compared to interaction energies calculated by full CI with the same basis set. As important as the formulas is the light our analysis throws on the meaning of order in intermolecular perturbation theory.