Structural innovations are typically hidden and often identified by means of a-priori economic reasoning. Under multivariate Gaussian model innovations there is no loss measure available to distinguish between particular identifying restrictions and rotations thereof. Based on a non Gaussian copula distribution framework, this paper proposes a loss statistic that can be used to discriminate between alternative identifying assumptions on the basis of higher order moment characteristics. The merits of Moment Targeted Structural Innovations are illustrated by means of Monte Carlo simulations and real data applications to bivariate systems of US stock prices and total factor productivity and of international breakeven inflation rates.