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Hyperbolic metasurfaces with unique dispersion properties can manipulate light–matter interactions according to the demands. However, due to their inherent physical properties, topological transitions (flat bands) exist only in the orthogonal directions, which greatly limit their application. Here, we unveil rich dispersion engineering and topological transitions in hyperbolic metasurfaces. Based on the effective medium theory, the rotation matrix is introduced into the dispersion relation to explain the distorted energy band diagrams, iso-frequency contours and higher-order multi-dipoles of the novel twisted metasurfaces, thereby forming multi-directional topological transitions and surface plasmon polariton propagation. Furthermore, we develop an integrated model to realize new dual-channel negative refraction and nondiffraction negative refraction. The phenomena observed in the experiments match well with the simulations, which proves that the designed metasurfaces make new types of negative refraction possible and will help to overcome the diffraction limit. The hyperbolic metasurfaces presented here exhibit exceptional capabilities for designing microscopes with a super lens at the molecular level, concealment of military aircraft, invisibility cloaks and other photonic devices with higher transmission efficiency.