Multilevel diversity coding was introduced in recent work by Roche (1992) and Yeung (1995). In a multilevel diversity coding system, an information source is encoded by a number of encoders. There is a set of decoders, partitioned into multiple levels, with each decoder having access to a certain subset of the encoders. The reconstructions of the source by decoders within the same level are identical and are subject to the same distortion criterion. Inspired by applications in computer communication and fault-tolerant data retrieval, we study a multilevel diversity coding problem with three levels for which the connectivity between the encoders and decoders is symmetrical. We obtain a single-letter characterization of the coding rate region and show that coding by superposition is optimal for this problem. Generalizing to a symmetrical problem with an arbitrary number of levels, we derive a tight lower bound on the coding rate sum.