A polarization-selective dual-band tunable terahertz metamaterial absorber is proposed and investigated. The absorber is composed of square split graphene rings, a SiO2 dielectric layer, and a Au reflection layer. The finite-difference time-domain simulation results show that the absorber can achieve dual-band high-efficiency absorption under different polarization light incidence. When the incident light is x-polarized, the absorption rates at 7.86 and 12.63THz are 97.9% and 91.2%, respectively; when the incident light is y-polarized, the absorption rates at 6.30 and 10.52THz are 94.1% and 93.2%, respectively. By changing the Fermi energy of graphene, the wavelengths of the dual-band absorption peaks of the two polarizations can be tuned. In addition, the effects of the thickness of the dielectric layer and the physical parameters of the graphene split ring on the resonant absorption peaks are investigated. Because dual-band high absorption can be generated in both polarization states, this absorber has important potential applications in terahertz polarization imaging, terahertz sensing, selective spectral detection, and polarization multiplexing.