Traditionally, optimizing compilers attempt to improve the performance of programs by applying source to source transformations, such as loop interchange, loop skewing and loop distribution. Each of these transformations has its own special legality checks and transformation rules which make it hard to analyze or predict the effects of compositions of these transformations. To overcome these problems we have developed a framework for unifying iteration reordering transformations. The framework is based on the idea that all reordering transformation can be represented as a mapping from the original iteration space to a new iteration space. The framework is designed to provide a uniform way to represent and reason about transformations. An optimizing compiler would use our framework by finding a mapping that both corresponds to a legal transformation and produces efficient code. We present the mapping selection problem as a search problem by decomposing it into a sequence of smaller choices. We then characterize the set of all legal mappings by defining an implicit search tree. (Also cross-referenced as UMIACS-TR-94-71)