The goal of this article is to identify and understand the fundamental role of spatial symmetries in the emergence of undulatory swimming using an anguilliform robot. Here, the local torque at the joints of the robot is controlled by a chain of oscillators forming a central pattern generator (CPG). By implementing a symmetric CPG with respect to the transverse plane, motor activation waves are inhibited, preventing the emergence of undulatory swimming and resulting in an oscillatory gait. We show experimentally that the swimmer can recover from the traveling wave inhibition by using distributed fluid force feedback to modulate the phase dynamics of each oscillator. This transition from oscillatory to undulating swimming is characterized by a symmetry breaking in the CPG and the body dynamics. By studying the stability of the oscillator chain, we show that the sensory feedback produces a frequency detuning gradient along the CPG chain while preserving its stability. To explain the origin of the instability, we introduce a toy model where the couplings between the dynamics of the oscillators and the body deformation reinforce the symmetry breaking.