  The notion of Lyndon word and Lyndon factorization has shown to have
unexpected applications in theory as well in developing novel algorithms on
words. A counterpart to these notions are those of inverse Lyndon word and
inverse Lyndon factorization. Differently from the Lyndon words, the inverse
Lyndon words may be bordered. The relationship between the two factorizations
is related to the inverse lexicographic ordering, and has only been recently
explored. More precisely, a main open question is how to get an inverse Lyndon
factorization from a classical Lyndon factorization under the inverse
lexicographic ordering, named CFLin. In this paper we reveal a strong
connection between these two factorizations where the border plays a relevant
role. More precisely, we show two main results. We say that a factorization has
the border property if a nonempty border of a factor cannot be a prefix of the
next factor. First we show that there exists a unique inverse Lyndon
factorization having the border property. Then we show that this unique
factorization with the border property is the so-called canonical inverse
Lyndon factorization, named ICFL. By showing that ICFL is obtained by
compacting factors of the Lyndon factorization over the inverse lexicographic
ordering, we provide a linear time algorithm for computing ICFL from CFLin.
