Wasserstein distributionally robust optimization (DRO), a leading paradigm in data-driven decision-making, requires evaluating worst-case risk over a high-dimensional Wasserstein ball. We study when this worst-case evaluation admits an exact reduction to a one-dimensional formulation, in the sense that it can be carried out over a one-dimensional Wasserstein ball centered at the projected reference distribution. We refer to this property as projection equivalence. We investigate projection equivalence across several classes of risk functionals. Starting from general law-invariant risk functionals and progressing through monotone risk functionals, coherent risk measures, and further specialized subclasses, we provide a complete characterization by giving necessary and sufficient conditions on the loss function under which projection equivalence holds. Beyond simplifying worst-case risk evaluation, our characterization also identifies when the worst-case problem admits an exact regularization reformulation, substantially extending previously known results. Applications to distributionally robust chance-constrained programs and classification problems are presented.