In this work we propose a new family of arbitrary-order mixed methods for anisotropic heterogeneous diffusion on general polyhedral meshes. A key ingredient is the choice of flux degrees of freedom, which allows one to define a discrete divergence operator that satisfies the usual commuting diagram property. Based on this choice and on the discrete divergence operator, we define a flux reconstruction with suitable consistency and stability properties. The flux reconstruction and the discrete divergence operator are then used to define the discrete counterparts of the bilinear forms that appear in the continuous mixed formulation. A convergence analysis in the energy norm is carried out, and a supercloseness result for the L2-norm of the potential is proved. Several variations are considered, and the link with existing methods in the lowest-order case is discussed. Finally, the most relevant implementation issues are discussed, and some numerical tests are presented.