We consider the idealized setting of gradient flow on the population risk for infinitely wide two-layer ReLU neural networks (without bias), and study the effect of symmetries on the learned parameters and predictors. We first describe a general class of symmetries which, when satisfied by the target function f* and the input distribution, are preserved by the dynamics. We then study more specific cases. When f* is odd, we show that the dynamics of the predictor reduces to that of a (non -linearly parameterized) linear predictor, and its exponential convergence can be guaranteed. When f* has a low-dimensional structure, we prove that the gradient flow PDE reduces to a lower-dimensional PDE. Furthermore, we present informal and numerical arguments that suggest that the input neurons align with the lower-dimensional structure of the problem.