  In this work it is proved that the Moore-Penrose inverse of the product of
$n$-doubly commuting regular $C^*$-algebra elements obeys the so-called reverse
order law. Conversely, conditions regarding the reverse order law of the
Moore-Penrose inverse are given in order to characterize when $n$-regular
elements doubly commute. Furthermore, applications of the main results of this
article to normal $C^*$-algebra elements, to Hilbert space operators and to
Calkin algebras will be considered.
