Cyclic structure and dynamics are of great interest in both the fields of stochastic process and non-equilibrium statistical physics.In this talk,we present a new symmetry of the linear Brownian motion named as the quasi-time-reversal invariance.It turns out that such an invariance of the Brownian motion is the key to prove an equality for diffusion processes on the circle named as the cycle symmetry,which says that the distributions of the forming times of the forward and backward cycles,under the condition that the corresponding cycle is formed earlier than the other,are exactly the same.We show that the cycle symmetry leads to the large deviation principle of the sample circulations of diffusion processes on the circle,in which the rate function has a highly non-obvious symmetry.Further extensions and applications are also discussed,especially the fluctuation theorems for the sample net circulation and the sample entropy production rate.