There are a number of recognized Rex breeds. But the common factor between them is the unique aspect of the fur. There have been a number of naturally occurring genetic mutations for these cats that gives each breed a distinctive wavy or curly fur.
The actual length and weight of that fur is not important, it can be long or short, thick or light. The particular mutated gene is the determining factor. The ancestors of the cats are not as important as the genetic material used for establishing the lines. These can be from any breed.
The specific gene also can spontaneously appear in a number of more standard breeds as well. The overall categorical name originates from the same genetic fur variant found in some of the rabbits available. This same trait the curly or wavy fur is also found in a number of other small animals, such as hamsters and Guinea pigs, as well.
At the present time there are four recognized breeds, all exhibiting the effect is of a separate and distinctive gene, but with little other similarity in form and feature for the breeds. The Cornish is slender, elegant in its appearance. On the opposite end of the spectrum is the Selkirk breed which exhibits a heavier boned body that is larger and more blocky.
At the present time there are also a number of other developing distinctive breeds. Since the genetic mutation seems to be independent of breeds, these are to be expected. As time goes by, some may even become established as separate: German, Tasman Group and Skookum are examples of the ongoing divisions.