Polaritons, i.e., hybrid quasi-particles of light and matter resonances, have been extensively investigated due to their potential to enhance light-matter interactions. Although polaritonic applications thrive in the mid-infrared range, their extension to the terahertz (THz) range remains limited. Here, we present paratellurite (α-TeO2) nanowires, a versatile material acting as a platform for different types of phonon polaritons. Utilizing synchrotron infrared nanospectroscopy from 10 to 24 THz, we uncover the polaritonic properties of α-TeO2 nanowires, showcasing their dual functionality as both a Fabry-Pérot cavity and a waveguide for surface phonon polaritons. Furthermore, near-field measurements with a free-electron laser as a THz source reveal a localized optical contrast down to 5.5 THz, an indication of hyperbolic bands. Our findings complement the repertoire of polaritonic materials, with significant implications for advancing THz technologies.