We measure the inflation-unemployment tradeoff associated with monetary easing and tightening, during booms and recessions, using a novel nonlinear Proxy-SVAR approach. We find evidence of significant nonlinearities for the U.S. economy (1973:M1 - 2019:M6): stimulating economic activity during recessions is associated with minimal costs in terms of inflation, and reducing inflation during booms delivers small costs in terms of unemployment. Overall, these results provide support for countercyclical monetary policies, in contrast with what predicted by a flat Phillips curve, or previous studies on nonlinear effects of monetary policy. The presence of an asymmetric monetary tradeoff can be rationalized by a simple model with downward nominal wage rigidity, which is also used to assess the validity of our empirical approach.