This work presents a rigorous framework based on coarse-graining to analyze highly compressible turbulence. We show how the requirement that viscous effects on the dynamics of large-scale momentum and kinetic energy be negligible ---an inviscid criterion--- naturally supports a density weighted coarse-graining of the velocity field. Such a coarse-graining method is already known in the literature as Favre filtering; however its use has been primarily motivated by appealing modeling properties rather than underlying physical considerations. We also prove that kinetic energy injection can be localized to the largest scales by proper stirring, and argue that stirring with an external acceleration field rather than a body force would yield a longer inertial range in simulations. We then discuss the special case of buoyancy-driven flows subject to a spatially-uniform gravitational field. We conclude that a range of scales can exist over which the mean kinetic energy budget is dominated by inertial processes and is immune from contributions due to molecular viscosity and external stirring. ; Comment: 31 pages, 1 figure, to appear in Physica D