We designed a cut-wire-pair metasurface that works as a transparent terahertz half-wave plate, by matching the electric and magnetic resonances of the structure. Due to the impedance matching nature of the resonances, a large transmission phase shift between the orthogonal polarizations was achieved, while permitting a high transmission. The electric and magnetic responses of the proposed structure were confirmed by evaluating the electric admittance and magnetic impedance. The structure was fabricated on a flexible film and its helicity-conversion function in the terahertz frequency range was experimentally demonstrated. The thickness of the device is less than 1/10 of the working vacuum wavelength, and a high amplitude helicity conversion rate over 80% was achieved. Finally, using simulations, we demonstrate the feasibility of the gradually rotating cut-wire-pair array in terahertz wave-front control.