The cross-coupling between strain and velocity in acoustic materials, known as Willis coupling, has recently been receiving increasing attention within the broad acoustics community. Willis coupling can provide a new degree of freedom to control sound propagation, which has been enabling several novel applications. In this work, based on the general constitutive relations of Willis media and the acoustic Poynting theorem, we study the constraints stemming from fundamental symmetries – parity, time-reversal, reciprocity and energy conservation – in these media. The wave features, such as wave-vectors, wave impedance and orthogonality are also generally investigated. In addition, we put forward a nonlocal model that unveils the relation between Willis media and nonlocal materials, and how Willis phenomena stem from weak forms of nonlocality.