A symmetric Banach space (SBS) is one whose dual ball is a bounded symmetric domain. It is a deep fundamental result that a contractively complemented subspace of an SBS is linearly isometric to another SBS. We prove the converse: if a subspace of a SBS is linearly isometric to an SBS, then that subspace is contractively complemented. Examples of SBSs are L^1-spaces and preduals of von Neumann algebras. (Joint work with B. Russo)