Efficiency and symmetric treatment of agents are the primary goals of resource allocation in environments without transfers. Focusing on ordinal mechanisms in which no small group of agents can substantially change the allocations of others, we show that all asymptotically efficient, symmetric, and asymptotically strategy-proof mechanisms lead to the same allocations in large markets. In particular, many mechanisms - both well-known and newly developed - are allocationally equivalent. This equivalence is consistent with prior empirical findings that different mechanisms lead to similar allocations in school choice. We also show that uniform randomizations over deterministic efficient mechanisms are asymptotically efficient.