We introduce the cycling normal form, a new normal form for the dual braid monoid. It is based on a natural operation called the phi-splitting, which expresses every dual n-braid in terms of a finite sequence of braids of dual (n-1)-braids. We deduce a complete description of the Dehornoy ordering of the dual braid monoids: via the phi-splitting, the ordering of dual n-braids is a lexicographical extension of the ordering of dual (n-1)-braids. We deduce a new proof for the existence of the braid ordering, and determine the order-type of the braid ordering of dual n-braids.