We study low-order reconstruction operators on polyhedral meshes, providing a unified framework for degrees of freedom attached to vertices, edges, faces, and cells. We present two equivalent sets of design properties and draw links with the literature. In particular, the two-level construction based on a P0-consistent and a stabilization part provides a systematic way of designing these operators. We present a simple example of piecewise constant reconstruction in each mesh cell, relying on geometric identities to fulfill the design properties on polyhedral meshes. Finally, we use these reconstruction operators to define a Hodge inner product and build Compatible Discrete Operator schemes, and we test the influence of the reconstruction operators in terms of accuracy and computational efficiency on an anisotropic diffusion problem.