Consider the object allocation (one-sided matching) model of Shapley and Scarf (1974). When final allocations are observed but agents' preferences are unknown, when might the allocation be in the core? This is a one-sided analogue of the model in Echenique, Lee, Shum, and Yenmez (2013). I build a model in which the strict core is testable -- an allocation is "rationalizable" if there is a preference profile putting it in the core. In this manner, I develop a theory of the revealed preferences of one-sided matching. I study rationalizability in both non-transferrable and transferrable utility settings. In the non-transferrable utility setting, an allocation is rationalizable if and only if: whenever agents with the same preferences are in the same potential trading cycle, they receive the same allocation. In the transferrable utility setting, an allocation is rationalizable if and only if: there exists a price vector supporting the allocation as a competitive equilibrium
or equivalently, it satisfies a cyclic monotonicity condition. The proofs leverage simple graph theory and combinatorial optimization and tie together classic theories of consumer demand revealed preferences and competitive equilibrium.