  A divide on an orientable 2-orbifold gives rise to a fibration of the unit
tangent bundle to the orbifold.We characterize the corresponding monodromies as
exactly the products of a left-veering horizontal and a right-veering vertical
antitwist with respect to a cylinder decomposition, where the notion of an
antitwist is an orientation-reversing analogue of a multitwist. Many divide
monodromies are pseudo-Anosov and we give plenty of examples.In particular, we
show that there exist divide monodromies with stretch factor arbitrarily close
to one, and give an example none of whose powers can be obtained by Penner's or
Thurston's construction of pseudo-Anosov mapping classes.As a side product, we
also get a new combinatorial construction of pseudo-Anosov mapping classes in
terms of products of antitwists.
