We exploit inequality restrictions on higher-order moments of the distribution of structural shocks to sharpen their identification. We show that these constraints can be treated as necessary conditions and used to shrink the set of admissible rotations. We illustrate the usefulness of this approach showing, by simulations, how it can dramatically improve the identification of monetary policy shocks when combined with widely used sign-restriction schemes. We then apply our methodology to two empirical questions: the effects of monetary policy shocks in the U.S. and the effects of sovereign bond spread shocks in the euro area. In both cases, using higher-moment restrictions significantly sharpens identification. After a shock to euro area government bond spreads, monetary policy quickly turns expansionary, corporate borrowing conditions worsen on impact, the real economy and the labor market of the euro area contract appreciably, and returns on German government bonds fall, likely reflecting investors’ flight to quality.