Orientation of a boundary manifold

This is a minor point, but worth note:
- there are a priori 4 natural choices for how to orient a boundary manifold:
  extend the orientation on the boundary via
  inner/outer pointing normal,
  either at the end or start (last/first).

This is actually a (torsor for the) vier-group of possibilities,
and the orientations change via:
 -1    (out/in)
(-1)^n (last/first: switch sign for odd, but not even)
[yes, it's a canonically split vier-group]

The correct orientation is the one that makes Stoke's theorem work;
(as "d" is natural, and this is adjoint)
I believe that this is outer normal last.

Actually, be very very careful:
when taking the dual of a chain complex,
there's the question of how to dualize the map:
some may want to add in an alternating sign.

[The more naive answer is:
 inward normal last,
 so the upper half-plane (and half-space) induce
 the natural orientation on the line,
 but this is rather lame]