Clustering, order conditions, and languages of interval exchanges

For any interval exchange transformation T (standard or generalized), if we define a morphism $ϕ$ from the set of letters to the set of the return words of a word in the natural coding, respecting the lexicographical order, the word $ϕv$ clusters (for the Burrows-Wheeler transform) for the permutation of T if and only if the word v clusters for the permutation of the induced map of T on the cylinder $[w]$. When T is symmetrical, all such natural codings are rich languages, and this implies that the two orders above are the same if w is a palindrome. Finally, we generalize the result, proved by using the clustering of a word w, that $ww$ is produced by a generalized interval exchange transformation if and only if $ww$ is produced by a standard interval exchange transformation, to non-clustering w: in the symmetric case, w is produced by a generalized interval exchange transformation if and only if w is produced by a standard interval exchange transformation.