It was proved by Gabber in the early 1980's that R\Psi commutes with duality, and that R\Phi preserves perversity up to shift. It had been in the folklore since then that this last result was in fact a consequence of a finer one, namely the compatibility of R\Phi with duality. In this talk I'll give a proof of this, using a method explained to me by A. Beilinson.