  Perfectly clustering words are one of many possible generalizations of
Christoffel words. In this article, we propose a factorization of a perfectly
clustering word on a $n$ letters alphabet into a product of $n-1$ palindromes
with a letter between each of them. This factorization allows us to generalize
two combinatorial characterization of Christoffel words due to Pirillo (1999)
and de Luca and Mignosi (1994).
