We devise mixed methods for heterogeneous anisotropic diffusion problems supporting general polyhedral meshes. For a polynomial degree $k\ge 0$, we use as potential degrees of freedom the polynomials of degree at most $k$ inside each mesh cell, whereas for the flux we use both polynomials of degree at most $k$ for the normal component on each face and fluxes of polynomials of degree at most $k$ inside each cell. The method relies on three ideas: a flux reconstruction obtained by solving independent local problems inside each mesh cell, a discrete divergence operator with a suitable commuting property, and a stabilization enjoying the same approximation properties as the flux reconstruction. Two static condensation strategies are proposed to reduce the size of the global problem, and links to existing methods are discussed. We carry out a full convergence analysis yielding flux-error estimates of order $(k+1)$ and $L^2$-potential estimates of order $(k+2)$ if elliptic regularity holds. Numerical examples confirm the theoretical results.