Twisted two-dimensional bilayer anisotropy materials exhibit many exotic physical phenomena. Manipulating the “twist angle” between the two layers enables the hybridization phenomenon of polaritons, resulting in fine control of the dispersion engineering of the polaritons in these structures. Here, combined with the hybridization phenomenon of anisotropy polaritons, we study theoretically the near-field radiative heat transfer (NFRHT) between two twisted hyperbolic systems. These two twisted hyperbolic systems are mirror images of each other. Each twisted hyperbolic system is composed of two graphene gratings, where there is an angle φ between these two graphene gratings. By analyzing the photonic transmission coefficient as well as the plasmon dispersion relation of the twisted hyperbolic system, we prove the enhancement effect of the topological transitions of the surface state at a special angle [from open (hyperbolic) to closed (elliptical) contours] on radiative heat transfer. Meanwhile the role of the thickness of dielectric spacer and vacuum gap on the manipulating the topological transitions of the surface state and the NFRHT are also discussed. We predict the hysteresis effect of topological transitions at a larger vacuum gap, and demonstrate that as the thickness of the dielectric spacer increases, the transition from the enhancement effect of heat transfer caused by the twisted hyperbolic system to a suppression.