In studies of the dynamics of a kicked pendulum, webs along which stochastic diffusion takes place, have been observed to occur at resonance. The author deals with stochastic webs with crystallographic symmetries. Via an analysis of the (reversing) symmetries of the equations of motion, symmetry properties of stochastic webs are revealed. Furthermore, the threshold for stochastic diffusion is discussed in relation to the (reversing) symmetries of the equations of motion.