A new approach, called Extreme-PLS, is proposed for dimension reduction in regression and adapted to distribution tails. The goal is to find linear combinations of predictors that best explain the extreme values of the response variable by maximizing the associated covariance. This adaptation of the PLS estimator to the extreme-value framework is achieved in the context of a non-linear inverse regression model. In practice, it allows quantifying the effect of the covariates on the extreme values of the response variable in a simple and interpretable way. Moreover, it should yield improved results for most estimators dealing with conditional extreme values thanks to the dimension reduction achieved in the projection step. From the theoretical point of view, the asymptotic normality of the Extreme-PLS estimator is established under a heavy tail assumption but without recourse to linearity or independence assumptions. The performance of the method is assessed on simulated data. Finally, the Extreme-PLS approach is used to analyze the influence of various parameters on extreme cereal yields collected on French farms.