For over 20 years the data clustering properties and applications of the efficient Burrows–Wheeler transform have been researched. Lexicographic suffix-sorting is induced during the transformation, and more recently a new direction has considered alternative ordering strategies for suffix arrays and thus the transforms. In this survey we look at these distinctly ordered bijective and linear transforms. For arbitrary alphabets we discuss the V-BWT derived from V-order and the D-BWT based on lex-extension order. The binary case yields a pair of transforms, the binary Rouen B-BWT, defined using binary block order. Lyndon words are relevant to implementing the original transform; the new transforms are defined for analogous structures: V-words, indeterminate Lyndon words, and B-words, respectively. There is plenty of scope for further non-lexicographic transforms as indicated in the conclusion.