This paper investigates the oil market reaction to its fundamental shocks: supply, aggregate demand and oil-specific demand in different regimes characterised by high versus low uncertainty in the market. We do so by first proposing a novel oil uncertainty index that is measured by the conditional volatility of the unpredictable component of oil prices. Then, we employ a nonlinear model to show that the structural oil market shocks have asymmetric effects. For instance, in relation to real economic activity, we find that both supply shocks and oil-specific demand shocks have negligible impacts in periods of low oil price uncertainty but sizeable effects in periods of high oil price uncertainty. Our model also enables us to evaluate the hypothesis that real economic activity responds asymmetrically to unexpected increases and decreases in oil prices driven by supply and specific demand shocks. We find that the effects of oil supply shocks are asymmetric but oil specific demand shocks are not, which indicates that the (a)symmetric oil market reaction depends on the underlying market shocks.