The (Heisenberg) Weyl algebra is the free algebra with two generators D and U and single relation DU−UD = 1. As a consequence of this relation, certain monomials are equal, such as DUUD and UDDU. We characterize all such equalities over a field of characteristic 0, describing them in several ways, including operationally (by a combinatorial equivalence relation generated by certain moves) and computationally (through lattice path invariants). We also enumerate the equivalence classes and several variants thereof and discuss possible extensions to other algebras.