In theories of the overall mechanical behaviour of polycrystals, composites, or other heterogeneous media, the transition from microscopic to macroscopic levels depends on finding connexions between suitably defined macro-variables and volume averages of microfields over a representative sample. Some connexions are established here for unrestricted deformation and internal rotations, without regard to constitutive properties. General measures of stress and finite strain are considered, together with objective fluxes of stress. The comparative advantages of such variables are assessed, more especially in relation to the averaging of tensor products. A bilinear differential form, akin to Poincare's integral invariant in Hamiltonian dynamics, plays an important role in the analysis.