Decomposing inflation into core and non-core components (e.g., energy) sheds new light on the nature of inflation risk and risk premia. While stocks have insignificant exposure to headline inflation in the U.S., their core inflation betas are negative and energy betas are positive. Conventional inflation hedges such as currencies and commodities only hedge against energy inflation risk but not the core. These hedging properties are reflected in the prices of inflation risks: only core inflation carries a negative risk premium and its magnitude is consistent both within and across asset classes, whereas the price of energy inflation risk is indistinguishable from zero. The relative contribution of core and energy inflation varies over time, which helps explain why the correlation between stock and bond returns appears to switch sign in the data. We develop a two-sector New Keynesian model to account for these facts.