  We establish the equivalence between the Multiscale Hybrid-Mixed (MHM) and
the Multiscale Hybrid High-Order (MsHHO) methods for a variable diffusion
problem with piecewise polynomial source term. Under the idealized assumption
that the local problems defining the multiscale basis functions are exactly
solved, we prove that the equivalence holds for general polytopal (coarse)
meshes and arbitrary approximation orders. We also leverage the interchange of
properties to perform a unified convergence analysis, as well as to improve on
both methods.
