This paper examines the dependence structure and risk spillovers between oil prices and exchange rates in both oil-exporting and oil-importing countries. Using a flexible dependence switching copula model, we analyze both positive and negative dependence and transitions between the dependence regimes. Additionally, we investigate the directional risk spillovers between oil and currency markets in both their downsides and upsides. Based on empirical data from 1999 to 2024 for major oil-exporting and oil-importing countries, we find that oil price-currency dependence is predominantly positive for oil-exporting countries, with infrequent transitions, but mainly negative for oil-importing countries, with frequent transitions between the two dependence regimes. These transitions often occur around crisis or war times. Furthermore, we observe that during downturns in the oil market, tail dependence between oil prices and currencies becomes more pronounced than during upturns. Our results indicate the presence of risk spillovers between oil and currency markets, with the downside spillover effects outweighing the upside ones. Moreover, we find that risk spillover is stronger from oil markets to currency markets than the reverse direction. These insights substantially enrich the existing literature and would offer valuable implications for effective risk management strategies and policymaking.