We design and analyze an approximation method for advection-diffusion-reaction equa-tions where the (generalized) degrees of freedom are polynomials of order k>=0 at mesh faces. The method hinges on local discrete reconstruction operators for the diffusive and advective derivatives and a weak enforcement of boundary conditions. Fairly general meshes with poly-topal and nonmatching cells are supported. Arbitrary polynomial orders can be considered, including the case k=0 which is closely related to Mimetic Finite Difference/Mixed-Hybrid Finite Volume methods. The error analysis covers the full range of Péclet numbers, including the delicate case of local degeneracy where diffusion vanishes on a strict subset of the domain. Computational costs remain moderate since the use of face unknowns leads to a compact stencil with reduced communications. Numerical results are presented.