In this paper, we proposed a dual-polarization ultra-wideband metasurface with half-reflection and half-transmission based on a double-split-ring (DSR) structure operating in the terahertz (THz) frequency range. The designed metasurface enables simultaneous control of circularly polarized (CP) and linearly polarized (LP) waves in both the reflection and transmission modes, covering an extensive THz frequency range. The unit-cell architecture of the metasurface consists of a periodic arrangement of the DSR structure made of metal, which is affixed to an ultra-thin dielectric substrate. By manipulating the size and rotation direction of the DSR structure, we achieve full phase coverage of 0-2π of the orthogonally polarized LP and CP waves across the frequency span of 0.3-1.2 THz, encompassing both transmission and reflection scenarios. The relative bandwidths of the orthogonal LP and CP waves for both reflection and transmission are significantly wide, reaching 86% and 120%, respectively, with an average amplitude of 0.45. Specifically, through numerical simulations, we demonstrated that the designed metasurfaces have ability to achieve THz beam deflection and vortex beam generation with simultaneously reflection and transmission for LP and CP waves. The proposed dual-polarization ultra-wideband metasurface holds great promise for various applications in the terahertz frequency range. These findings pave the way for the development of flexible and versatile THz devices with expanded capabilities, opening up new possibilities for wavefront manipulation in metasurfaces.