We investigate the very long-wavelength, global pattern of surface heat flux anomalies within the context of whole-mantle and layered-mantle anelastically compressible internal loading theories. Since the internal loading framework does not yield a direct estimate of the geotherm, we argue that accurate predictions for the surface heat flux may nevertheless be obtained by assuming that it is linearly related to the radial component of flow velocity at shallow depth in the mantle. The mantle convective circulation is assumed to be driven by density heterogeneity inferred from global seismic tomography models. Best results for the pattern of surface heat flux anomalies are obtained for models that significantly impede the circulation at a depth of 670 km. Total variance reductions of 60--65% (degree 1--5) are obtained when the viscosity profile includes a low-viscosity asthenosphere. Within the context of our modeling assumptions, however, whole-mantle circulation models provide best descriptions of the long-wavelength nonhydrostatic gravity data. In order to resolve the gravity-heat flux impasse that is revealed herein, we consider the possibility of modifying the a priori global seismic models employed in the calculations. We show that the rigidly layered-mantle internal loading theory is equivalent to a theory in which no explicit flow-blocking boundary condition is imposed at 670 km but in which the buoyancy field inferred from the a priori tomographic model is supplemented by flow-blocking heterogeneity in the form of an appropriately constrained sheet mass load. We develop a general mathematical formalism describing how the introduction of appropriately constrained sheet mass loads allows the exact reconciliation of a number of a priori constraints or hypotheses concerning the structure of the circulation. Using this formalism, we explore the extreme nonuniqueness that not only characterizes internal loading theory inferences of the depth profile of mantle viscosity but also inferences of the radial style of the circulation. On this basis, we suggest that great caution is warranted with respect to tomography-based inferences of mantle properties. Based on a viscosity profile whose depth dependence is close to that independently inferred within the context of postglacial rebound studies, we present plausible resolutions of the gravity-heat flux impasse effected either within the framework of whole-mantle or layered-mantle circulation models. ¿ 1998 American Geophysical Union