Given a scalar-valued discontinuous piecewise polynomial, a potential reconstruction is a piecewise polynomial that is trace continuous, i.e., H1-conforming. It is best obtained via a conforming finite element solution of local homogeneous Dirichlet problems on patches of elements sharing a mesh vertex, using the conforming finite element method. Similarly, given a vector-valued discontinuous piecewise polynomial not satisfying the target divergence, a flux reconstruction is a piecewise polynomial that is normal-trace continuous, i.e., H(div)-conforming, and has the target divergence. It is best obtained via local homogeneous Neumann problems on patches of elements, using the mixed finite element method. The concepts of potential and flux reconstructions are known to lead to guaranteed, locally efficient, and polynomial-degree-robust a posteriori error estimates. Such use is based on the Prager–Synge equality and we recall it here. In the remainder of the presentation, we show that potential and flux reconstructions also allow to obtain novel results in a priori error analysis. They actually allow to devise stable local commuting projectors that lead to p-robust equivalence of global-best approximation over the whole computational domain using a conforming finite element space with local- (elementwise-)best approximations without any continuity requirement along the interfaces, and without any constraint on the divergence. Therefrom, optimal hp approximation / a priori error estimates under minimal elementwise Sobolev regularity follow. The main tools are piecewise polynomial extension operators and p-stable decompositions.