Real world auctions are often restricted to being anonymous and nondiscriminatory (symmetric) due to practical or legal constraints. We examine the extent to which these constraints restrict the seller. We characterize the set of incentive compatible and individually rational allocation rules that can be implemented via a symmetric auction. An important implication is that the optimal (revenue-maximizing) auction can always be implemented symmetrically. We also characterize the set of outcomes implementable by a symmetric, ex-post individually rational mechanism. The optimal auction may not be implementable with this additional requirement.