A collaboration between DESY, PSI and CERN has developed and built an advanced modular X-band transverse deflection structure (TDS) system with the new feature of providing variable polarization of the deflecting force. The prototype of the novel X-band TDS, the polarizable $X$-band (PolariX) TDS, was fabricated at PSI following the high-precision tuning-free production process developed for the C-band Linac of the SwissFEL project. Bead-pull rf measurements were also performed at PSI to verify, in particular, that the polarization of the dipole fields does not have any rotation along the structure. The high-power test was performed at CERN and now the TDS is at DESY and has been installed in the FLASHForward beamline, where the first streaking experience with beam has been accomplished. We summarize in this paper the rf design of the TDS and its key components, such as the $X$-band pulse compressor, E-rotator, and phase shifter, the results of the bead-pull measurements and the high power test and finally the rf setup at DESY.