  We consider two shellings of the boundary of the hypercube equivalent if one
can be transformed into the other by an isometry of the cube. We observe that a
class of indecomposable permutations, bijectively equivalent to standard double
occurrence words, may be used to encode one representative from each
equivalence class of the shellings of the boundary of the hypercube. These
permutations thus encode the shelling types of the boundary of the hypercube.
We construct an adjacent transposition Gray code for this class of
permutations. Our result is a signed variant of King's result showing that
there is a transposition Gray code for indecomposable permutations.
