  We introduce a new type of discrete quantum walks, called vertex-face walks,
based on orientable embeddings. We first establish a spectral correspondence
between the transition matrix $U$ and the vertex-face incidence structure.
Using the incidence graph, we derive a formula for the principal logarithm of
$U^2$, and find conditions for its underlying digraph to be an oriented graph.
In particular, we show this happens if the vertex-face incidence structure
forms a partial geometric design. We also explore properties of vertex-face
walks on the covers of a graph. Finally, we study a non-classical behavior of
vertex-face walks.
