During 8--30 January 2004, a sequence of 68 UV images of Saturn's southern aurora was obtained by the Hubble Space Telescope (HST), coordinated for the first time with measurements of the upstream interplanetary conditions made by the Cassini spacecraft. Using the poleward edge of the observed aurora as a proxy for the open-closed field line boundary, the open flux content of the southern polar region has been estimated. It is found to range from ~15 to ~50 GWb during the interval, such a large variation providing evidence of a significant magnetospheric interaction with the solar wind, in particular with the interplanetary structures associated with corotating interaction regions (CIRs). The open flux is found to decline slowly during a rarefaction region in which the interplanetary magnetic field remained very weak, while decreasing sharply in association with the onset of CIR-related solar wind compressions. Such decreases are indicative of the dominating role of open flux closure in Saturn's tail during these intervals. Increases in open flux are found to occur in the higher-field compression regions after the onsets, and in a following rarefaction region of intermediate field strength. These increases are indicative of the dominating role of open flux production at Saturn's magnetopause during these intervals. The rate of open flux production has been estimated from the upstream interplanetary data using an empirical formula based on experience at Earth, with typical values varying from ~10 kV during the weak-field rarefaction region, to ~200 kV during the strong-field compression. These values have been integrated over time between individual HST image sets to estimate the total open flux produced during these intervals. Comparison with the changes in open flux obtained from the auroral images then allows us to estimate the amount of open flux closed during these intervals, and hence the averaged tail reconnection rates. Intermittent intervals of tail reconnection at rates of ~30--60 kV are inferred in rarefaction regions, while compression regions are characterised by rates of ~100--200 kV, these values representing averages over the ~2-day intervals between HST image sequences. The forms of the aurorae observed are also discussed in relation to the deduced voltage values.