Abstract: We introduce an operator depending on the "transverse velocity'' r that describes the effect of hadron masses on the leading 1/Q power correction to event-shape observables. Here, Q is the scale of the hard collision. This work builds on earlier studies of mass effects by Salam and Wicke [J. High Energy Phys. 05 (2001) 061] and of operators by Lee and Sterman [Phys. Rev. D 75, 014022 (2007)]. Despite the fact that different event shapes have different hadron mass dependence, we provide a simple method to identify universality classes of event shapes whose power corrections depend on a common nonperturbative parameter. We also develop an operator basis to show that at a fixed value of Q, the power corrections for many classic observables can be determined by two independent nonperturbative matrix elements at the 10% level. We compute the anomalous dimension of the transverse velocity operator, which is multiplicative in r and causes the power correction to exhibit nontrivial dependence on Q. The existence of universality classes and the relevance of anomalous dimensions are reproduced by the hadronization models in Pythia 8 and Herwig++, though the two programs differ in the values of their low-energy matrix elements.