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.