We propose and analyze a novel symmetric exponential wave integrator (sEWI) for the nonlinear Schr\"odinger equation (NLSE) with low regularity potential and nonlinearity. The sEWI is explicit and stable under a time step size restriction independent of the mesh size. We rigorously establish error estimates of the sEWI under various regularity assumptions on potential and nonlinearity. Extensive numerical results are reported to confirm our error estimates and to demonstrate the superiority of the sEWI, including much weaker regularity requirements on potential and nonlinearity and excellent long-time behavior with near-conservation of mass and energy.