  Almost two decades ago Zapponi introduced a notion of orientability of a
clean dessin d'enfant, based on an orientation of the embedded bipartite graph.
We extend this concept, which we call Z-orientability to distinguish it from
the traditional topological definition, to the wider context of all dessins,
and we use it to define a concept of twist orientability, which also takes
account of the Z-orientability properties of those dessins obtained by
permuting the roles of white and black vertices and face-centres. We observe
that these properties are Galois-invariant, and we study the extent to which
they are determined by the standard invariants such as the passport and the
monodromy and automorphism groups. We find that in general they are independent
of these invariants, but in the case of regular dessins they are determined by
the monodromy group.
