The mean density and mean moment of inertia are remotely accessible geodetic parameters which provide two integral constraints on the radial density profile of Mars. The mean density is presently well known, but until the axial precession rate is better determined, the moment of inertia will remain uncertain. Values anywhere in the interval from 0.345 MR2 to 0.365 MR2 can be obtained from the observed gravitational field, depending on how that field is partitioned into hydrostatic and nonhydrostatic components. Even if the moment were accurately known, a considerable degree of nonuniqueness would still remain concerning the density and composition. This nonuniqueness is illustrated by construction of a family of orthogonal polynomials in the normalized radius which make no contribution to either the mean density or the mean inertial moment. If the mean moment is near to the lower end of its plausible range, the nonuniqueness in density and composition is considerably diminished. In order to avoid unrealistically high densities in the deep interior, the Martian mantle must have a density appreciably lower than the terrestrial mantle and must consequently be significantly depleted in iron. On the other hand, if the mean moment is near the upper end of its plausible range, a much broader range of density profiles can be accommodated, including (but not restricted to) iron enriched mantles.