Longitudinal electron-beam diagnostics play a critical role in the operation and control of x-ray free-electron lasers, which rely on parameters such as the current profile, the longitudinal phase space, or the slice emittance of the particle distribution. On the one hand, the femtosecond-scale electron bunches produced at these facilities impose stringent requirements on the resolution achievable with the diagnostics. On the other, research and development of novel accelerator technologies such as beam-driven plasma-wakefield accelerators (PWFA) demand unprecedented capabilities to resolve the centroid offsets in the full transverse plane along the longitudinal bunch coordinate. We present the beam-based commissioning of an advanced X-band transverse-deflection rf structure (TDS) system with the new feature of providing variable polarization of the deflecting force: the PolariX-TDS. By means of a comprehensive campaign of measurements conducted with the prototype, key parameters of the rf performance of the system are validated and a phase-space characterization of an electron bunch is accomplished with a time resolution of 3.3 fs. Furthermore, an analysis of second-order effects induced on the bunch from its passage through the PolariX-TDS is presented.