We present a novel mean-field theory for balanced neuronal networks with arbitrary levels of symmetry in the synaptic connectivity. The theory determines the fixed point of the network dynamics and the conditions for its stability. The fixed point becomes unstable by increasing the synaptic gain beyond a critical value that depends on the level of symmetry. Beyond this critical gain, for positive levels of symmetry, we find a previously unreported phase. In this phase, the dynamical landscape is dominated by a large number of marginally-stable fixed points. As a result, the network dynamics exhibit non-exponential relaxation and ergodicity is broken. We discuss the relevance of such a glassy phase for understanding dynamical and computational aspects of cortical operation.