In this paper, we theoretically and numerically demonstrate a polarization-controlled and symmetry-dependent multiple plasmon-induced transparency (PIT) in a graphene-based metasurface. The unit cell of metasurface is composed of two reversely placed U-shaped graphene nanostructures and a rectangular graphene ring stacking on a dielectric substrate. By adjusting the polarization of incident light, the number of transparency windows can be actively modulated between 1 and 2 when the nanostructure keeps a geometrical symmetry with respect to the x-axis. Especially, when the rectangular graphene ring has a displacement along the y-direction, the number of transparency windows can be arbitrarily switched between 2 and 3. The operation mechanism behind the phenomena can be attributed to the near-field coupling and electromagnetic interaction between the bright modes excited in the unit of graphene resonators. Moreover, the electromagnetic simulations obtained by finite-difference time-domain (FDTD) method agree well with the theoretical results based on the coupled modes theory (CMT). In addition, as applications of the designed nanostructure, we also study the modulation degrees of amplitude, insertion loss and group index of transmission spectra for different Fermi energies, which demonstrates an excellent synchronous switch functionality and slow light effect at multiple frequencies. Our designed metasurface may have potential applications in mid-infrared optoelectronic devices, such as optical switches, modulators, and slow-light devices, etc.