To distinguish between clockwise and counterclockwise requires a system of axes. In a finite space, this is just a sequence of n reference boundaries in n dimensions. That any less would not suffice follows from the definition.

A simplex base selects one polyant of the simplex apex. The first of n reference boundaries selects the same or opposite polyant. The simplex is clockwise iff the same polyant is selected by base and reference. Removing the first reference boundary, recurse to find clockwise of the base, xoring with clockwise of the simplex. Then each segment goes from the segment apex to the segment base iff the segment is clockwise.

It remains to prove that any sequence of simplex boundaries that leads to the same vertex leads to the same clockwiseness. And that the counterclockwise endpoint of any segment is the clockwise endpoint of another. And that migrations of the simplex’s boundaries that do not migrate the simplex’s inside do not change clockwiseness of the segment endpoints. The proof of migration invariance would be interesting because it would validate a definition of migrate. Also prove that clockwiseness of a subregion of a simplex as the clockwiseness of the simplex is well defined.

It remains to prove that any sequence of simplex boundaries that leads to the same vertex leads to the same clockwiseness. And that the counterclockwise endpoint of any segment is the clockwise endpoint of another. And that migrations of the simplex’s boundaries that do not migrate the simplex’s inside do not change clockwiseness of the segment endpoints. The proof of migration invariance would be interesting because it would validate a definition of migrate. Also prove that clockwiseness of a subregion of a simplex as the clockwiseness of the simplex is well defined.