Introduced about thirty years ago in the field of Data Compression, the Burrows-Wheeler Transform (BWT) is a string transformation that, besides being a booster of the performance of memoryless compressors, plays a fundamental role in the design of efficient self-indexing compressed data structures. Finding other string transformations with the same remarkable properties of BWT has been a challenge for many researchers for a long time. Among the known BWT variants, the only one that has been recently shown to be a valid alternative to BWT is the Alternating BWT (ABWT), another invertible string transformation introduced about ten years ago in connection with a generalization of Lyndon words. In this paper, we introduce a whole class of new string transformations, called local orderings-based transformations, which have all the myriad virtues of BWT. We show that this new family is a special case of a much larger class of transformations, based on context adaptive alphabet orderings, that includes BWT and ABWT. Although all transformations support pattern search, we show that, in the general case, the transformations within our larger class may take quadratic time for inversion and pattern search. As a further result, we show that the local orderings-based transformations can be used for the construction of the recently introduced r-index, which makes them suitable also for highly repetitive collections. In this context, we consider the problem of finding, for a given string, the BWT variant that minimizes the number of runs in the transformed string, and we provide an algorithm solving this problem in linear time.