Broadband-tunable sources of circularly-polarized light are crucial in fields such as laser science, biomedicine and spectroscopy. Conventional sources rely on nonlinear wavelength conversion and polarization control using standard optical components, and are limited by the availability of suitably transparent crystals and glasses. Although gas-filled hollow-core photonic crystal fiber provides pressure-tunable dispersion, long well-controlled optical path-lengths, and high Raman conversion efficiency, it is unable to preserve circular polarization state, typically exhibiting weak linear birefringence. Here we report a revolutionary approach based on helically-twisted hollow-core photonic crystal fiber, which displays circular birefringence, thus robustly maintaining circular polarization state against external perturbations. This makes it possible to generate pure circularly-polarized Stokes and anti-Stokes signals by rotational Raman scattering in hydrogen. The polarization state of the frequency-shifted Raman bands can be continuously varied by tuning the gas pressure in the vicinity of the gain suppression point. The results pave the way to a new generation of compact and efficient fiber-based sources of broadband light with fully-controllable polarization state.