Planetary stratospheres are characterized by a subtle interplay between dynamics, radiation, and chemistry. Observations of Saturn's stratosphere revealed a semi-annual equatorial oscillation of temperature and hinted at an inter-hemispheric circulation of hydrocarbon species. Both the forcing mechanisms of the former and the existence of the latter have remained debated. Here we use a new troposphere-to-stratosphere Saturn global climate model to argue that those two open questions are intimately connected. Our Saturn climate model reproduces a stratospheric oscillation exhibiting the observed semi-annual period, amplitude, and downward propagation. In the same Saturn simulation, a prominent stratospheric summer-to-winter hemispheric circulation develops at solstices, controlled by both the seasonal radiative gradients and Rossby-wave pumping in the winter subsiding branch, analogous to Earth's Brewer-Dobson circulation. Furthermore, we show that Saturn's equatorial oscillation is driven by the seasonal variability of both the resolved planetary-scale wave activity and the inter-hemispheric circulation, akin to Earth's Semi-Annual Oscillation.