Carbonell-Nicolau and Llavador (forthcoming) extend the classic result of Jakobsson (1976) and Fellman (1976) according to which average-rate progressive, and only average-rate progressive income taxes, reduce income inequality to the case of endogenous income. There it is shown that marginal-rate progressivity in the sense of increasing marginal tax rates on income is necessary for tax structures to be inequality reducing, and necessary and sufficient conditions on the social utility function are identified under which progressive and only progressive taxes are inequality reducing. This paper takes a further step and furnishes conditions on primitives under which various subclasses of progressive taxes are inequality reducing. The main results in Carbonell-Nicolau and Llavador (forthcoming) are obtained as particular cases of the more general framework presented here. Restricting the set of taxes allows for larger classes of preferences consistent with inequality reducing income taxation. As an illustration of the results practical implications, we provide a precise characterization of the subclass of (progressive) taxes that are inequality reducing for some standard families of preferences.