Based on a dielectric loaded graphene plasmon waveguide (DLGPW), this study proposes and investigates a single dielectric loaded two-sheet graphene symmetric surface plasmon waveguide (DLTGSSPW). In the DLTGSSPW, the interaction between the surface plasmon polaritons (SPPs) in two graphene sheets induces the coupled SPP modes, i.e., the symmetric and anti-symmetric SPP modes. The effective index method and the finite element method are used to reveal that the effective mode refractive indexes, propagation losses, mode numbers, and electromagnetic fields of the coupled SPP modes are strongly dependent on the DLTGSSPW parameters, such as the incident wavelength and the width and height of a single dielectric strip. The coupled SPP modes are similar to the guided modes in a three-layer dielectric planar waveguide. In addition, when the single dielectric strip is sufficiently high, the symmetric and anti-symmetric SPPs degenerate into the uncouple SPP modes in the respective graphene sheets, and this structure can be considered as two independent DLGPWs. All the results about SPP waveguide may have some possible application in actively integrated optics.