Wireless communications based on reconfigurable intelligent surfaces (RIS) have upsurged as a key research topic in the communication-theoretic arena. Recently, the RIS concept has evolved into the so-called simultaneously transmitting and reflecting intelligent omni-surfaces (STAR-IOS, or simply IOS), which extend the RIS functionality by incorporating transmission (in addition to reflection) capabilities. To realize STAR-IOS, full and independent reconfiguration capabilities for both reflected and transmitted waves are crucially required. However, such full independent reconfiguration has thus far been hurdled by the intimate coupling between the transmission and reflection behaviour of the IOS elements. To overcome this challenge and realize the full potential of reconfigurable IOS-aided systems, we here advocate for the use of the polarization domain in the design and operation of STAR-IOS. Thanks to the polarization- dependent features of IOS elements, fully independent (reconfigurable) transmission and reflection modes can be delivered, thus bringing key performance improvements and opportunities for new communication scenarios.