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We study the constraining power of a high-precision measurement of the gravity field for Uranus and Neptune, as could be delivered by a low periapse orbiter. Our study is practical, assessing the possible deliverables and limitations of such a mission with respect to the structure of the planets. Our study is also academic, assessing in a general way the relative importance of the low order gravity, high order gravity, rotation rate, and moment of inertia (MOI) in constraining planetary structure. We attempt to explore all possible interior density structures of a planet that are consistent with hypothetical gravity data, via MCMC sampling of parameterized density profiles. When the gravity field is poorly known, as it is today, uncertainties in the rotation rate on the order of 10 minutes are unimportant, as they are interchangeable with uncertainties in the gravity coefficients. By the same token, when the gravity field is precisely determined the rotation rate must be known to comparable precision. When gravity and rotation are well known the MOI becomes well-constrained, limiting the usefulness of independent MOI determinations unless they are extraordinarily precise. For Uranus and Neptune, density profiles can be well-constrained. However, the non-uniqueness of the relative roles of H/He, watery volatiles, and rock in the deep interior will still persist with high-precision gravity data. Nevertheless, the locations and magnitudes (in pressure-space) of any large-scale composition gradient regions can likely be identified, offering a crucially better picture of the interiors of Uranus or Neptune.