Wasserstein distributionally robust optimization (DRO), a leading paradigm in data-driven decision-making, entails the evaluation of worst-case risk over a high-dimensional Wasserstein ball–a major computational burden. In this paper, we study when the worst-case risk problem admits an exact reduction to the evaluation of risk over a one-dimensional projected Wasserstein ball—a property we refer to as projection equivalence. This reduction depends on the class of risk functions used to evaluate risk: starting from the most general law-invariant risk functions and progressing through monotone risk functions, coherent risk measures, and further specialized classes, we provide a complete characterization–namely, necessary and sufficient conditions on the loss function under which projection equivalence holds. This not only simplifies the evaluation of worst-case risk but also enables a further characterization of cases in which the worst-case problem admits an exact regularization reformulation, significantly extending beyond previously known results. Applications to distributionally robust chance-constrained programs and classification problems are presented.