Metasurfaces, as a two-dimensional (2D) version of metamaterials, have drawn considerable attention for their revolutionary capability in manipulating the amplitude, phase, and polarization of light. As one of the most important types of metasurfaces, geometric metasurfaces provide a versatile platform for controlling optical phase distributions due to the geometric nature of the generated phase profile. However, it remains a great challenge to design geometric metasurfaces for realizing spin-switchable functionalities because the generated phase profile with the converted spin is reversed once the handedness of the incident beam is switched. Here, we propose and experimentally demonstrate chiral geometric metasurfaces based on intrinsically chiral plasmonic stepped nanoapertures with a simultaneously high circular dichroism in transmission (CDT) and large cross-polarization ratio (CPR) in transmitted light to exhibit spin-controlled wavefront shaping capabilities. The chiral geometric metasurfaces are constructed by merging two independently designed subarrays of the two enantiomers for the stepped nanoaperture. Under a certain incident handedness, the transmission from one subarray is allowed, while the transmission from the other subarray is strongly prohibited. The merged metasurface then only exhibits the transmitted signal with the phase profile of one subarray, which can be switched by changing the incident handedness. Based on the chiral geometric metasurface, both chiral metasurface holograms and the spin-dependent generation of hybrid-order Poincaré sphere beams are experimentally realized. Our approach promises further applications in spin-controlled metasurface devices for complex beam conversion, image processing, optical trapping, and optical communications.