B. Grebert, T. Kappeler Symmetries of the Nonlinear Schr\"odinger Equation (53K, Latex) ABSTRACT. Fundamental symmetries of the defocusing nonlinear Schr\"odinger equation are expressed in action-angle coordinates and characterized in terms of periodic and Dirichlet spectrum of the associated Zakharov-Shabat system. As a main application we prove a conjecture, raised by several experts in field, that the periodic spectrum is symmetric iff the sequence of gap lengths $(\gamma_k)_{k\in \mathbb {Z}}$ or, equivalently, the sequence of actions $(I_k)_{k\in \mathbb {Z}}$ is symmetric with respect to $k=0$.