  The notion of inverse Lyndon word is related to the classical notion of
Lyndon word. More precisely, inverse Lyndon words are all and only the nonempty
prefixes of the powers of the anti-Lyndon words, where an anti-Lyndon word with
respect to a lexicographical order is a classical Lyndon word with respect to
the inverse lexicographic order. Each word $w$ admits a factorization in
inverse Lyndon words, named the canonical inverse Lyndon factorization
$\ICFL(w)$, which maintains the main properties of the Lyndon factorization of
$w$. Although there is a huge literature on the Lyndon factorization, the
relation between the Lyndon factorization $\CFL_{in}$ with respect to the
inverse order and the canonical inverse Lyndon factorization $\ICFL$ has not
been thoroughly investigated. In this paper, we address this question and we
show how to obtain one factorization from the other via the notion of grouping.
This result naturally opens new insights in the investigation of the
relationship between $\ICFL$ and other notions, e.g., variants of Burrows
Wheeler Transform, as already done for the Lyndon factorization.
