  We develop a theory of minors for alternating dimaps --- orientably embedded
digraphs where, at each vertex, the incident edges (taken in the order given by
the embedding) are directed alternately into, and out of, the vertex. We show
that they are related by the triality relation of Tutte. They do not commute in
general, though do in many circumstances, and we characterise the situations
where they do. The relationship with triality is reminiscent of similar
relationships for binary functions, due to the author, so we characterise those
alternating dimaps which correspond to binary functions. We give a
characterisation of alternating dimaps of at most a given genus, using a finite
set of excluded minors. We also use the minor operations to define simple Tutte
invariants for alternating dimaps and characterise them. We establish a
connection with the Tutte polynomial, and pose the problem of characterising
universal Tutte-like invariants for alternating dimaps based on these minor
operations.
