The investigation of converged twisted beams with a helical phase structure has a remarkable impact on both fundamental physics and practical applications. Geometric metasurfaces consisting of individually orientated metal/dielectric meta-atoms provide an ultracompact platform for generating converged vortices. However, it is still challenging to simultaneously focus left-handed and right-handed circularly polarized incident beams with pure geometric phase modulation, which hinders the independent operation on topological charges between these two helical components. Here we propose and experimentally demonstrate an approach to design terahertz geometric metasurfaces that can generate helicity-independent converged vortices with homogeneous polarization states by the superposition of two orthogonal helical vortices with identical topological charges. Furthermore, the multiplexing of polarization-rotatable multiple vortices in multiple dimensions, i.e., in both longitudinal and transverse directions, and a vortex with an extended focal depth is confirmed by embedding polarization modulation into the geometric metasurfaces. The demonstrated approach provides a new way to simultaneously manipulate orthogonal helical components and expand the design dimension, enabling new applications of geometric metasurface devices in polarization optics, twisted-beam related image and edge detection, high capacity optical communication, and quantum information processing, to name a few.