We characterise the interaction between the solar wind and Saturn’s magnetosphere by evaluating the amount of ‘open’ magnetic flux connected to the solar wind. This is deduced from a large set of Hubble Space Telescope images of the ultraviolet aurora, using the poleward boundary of the main aurora as a proxy for the open-closed field line boundary in the ionosphere. The amount of open flux is found to be 10–50 GWb, with a mean of 35 GWb. The typical change in open flux between consecutive observations separated by 10–60 h is -5 or +7 GWb. These changes are a result of imbalance between open flux creation at the dayside magnetopause and its closure in the magnetotail. The 5 GWb typical decrease in open flux is consistent with in situ measurements of the flux transported following a reconnection event. Estimates of average, net reconnection rates are found to be typically a few tens of kV, with some extreme examples of unbalanced magnetopause or tail reconnection occurring at ~300 kV. The range of values determined suggest that Saturn’s magnetosphere does not generally achieve a steady state between flux opening at the magnetopause and flux closure in the magnetotail. The percentage of magnetic flux which is open in Saturn’s magnetosphere is similar to that measured at the Earth (2–11%), but the typical percentage that is closed between observations is significantly lower (13% compared to 40–70%). Therefore, open flux is usually closed in smaller (few GWb) events in Saturn’s magnetosphere. The exception to this behaviour is large, rapid flux closure events which are associated with solar wind compressions. While the rates of flux opening and closure should be equal over long timescales, they are evidently different on shorter (up to tens of hours) timescales. The relative independence of the magnetopause and tail reconnection rates can be attributed to the long loading timescales required to transport open field lines into the tail.