It is shown theoretically that the steady rotation of a container filled with a spatially uniform suspension of identical, otherwise neutrally buoyant, dipolar spherical particles in a Newtonian fluid, each containing an embedded gravitational dipole (due to an assumed mass inhomogeneity within the sphere), will give rise to an antisymmetric stress state on the length scale of the suspension. Simultaneously, an external torque must be continuously exerted on the container by some agency to maintain its steady rotation. These phenomena obtain despite the fact that the suspension-scale motion is trivially a rigid-body rotation. This result is interpreted within the general framework of classical structured continua/director/micropolar fluid theories, supplemented with a mass distribution polarization density field. For the case of concentrated suspensions of spheres whose centers are arrayed in a cubic lattice (but are otherwise free to rotate about axes through their centers), expressions are obtained for all of the parameters required for a complete characterization of the phenomena. Simple experiments are proposed for measuring the vortex viscosity of suspensions in an unequivocal manner, essentially free of the question of internal spin boundary conditions. In dilute suspensions, first-order wall effects upon the translational and rotational motions of isolated spherical particles are employed to interpret the heretofore unexplained macroscopic “swirling” motions observed in ferrofluid suspensions when subjected to a circularly rotating magnetic field. In particular, it is shown inter alia that this swirling motion (relative to the fixed container walls) is a manifestation of the existence of a suspension-scale translational slip velocity occurring at the container walls, arising from a combination of wall and external field effects. This translational slip, which—being a wall effect—is relatively small, is in addition to a much larger suspension-scale rotational slip velocity, whose existence is virtually independent of wall effects.