Stokes theorem for manifolds without orientation?

This is just more information on Jose's comment. Look at section 10 (pages 122 to 128) in:

Peter W. Michor: Topics in Differential Geometry. Graduate Studies in Mathematics, Vol. 93 American Mathematical Society, Providence, 2008. (pdf)

On the orientable double cover $\tilde M$ of $M$ (given by $\lbrace \xi\in Or(M): |\xi|=1\rbrace$ using 10.7) there are two kinds differential forms, namely the eigenspaces for $\pm 1$ under the action of the covering map. These are the "formes paire" and "formes impaire" on $M$ in the book of de Rham; for these you formulate Stokes' theorem directly on $M$, and it is the same as the Stokes' theorem on $\tilde M$. Note that you also have the corresponding double cover of the boundary.

This is mostly an elaboration on Peter Michor's answer, but I think more deserves to be said, including a major caveat. A reference which treats this material more fully and explicitly than his book cited above is Theodore Frankel's The Geometry of Physics.

From the "double-cover" perspective, I believe the usual differential forms are the ones in the $+1$ eigenspace of the covering map, which de Rahm calls paire ("even"). The ones in the $-1$ eigenspace ("odd forms" impaire) are called "pseudo-forms" by Frankel, although Frankel doesn't explicitly work in terms of the double cover. See section 2.8.e of Frankel for his definitions and section 3.4 for a discussion of integration, including a statement of Stokes' theorem for odd forms. (I had trouble finding such a statement in Michor's book...)

The caveat I want to mention is that like the familiar even forms, odd forms do require some orientation data to integrate over a submanifold (as in Stokes' Theorem). It's just different data. Whereas an even form is integrated over an oriented submanifold, an odd form is integrated over a transverse-oriented submanifold. That is, if $N^k \subset M^n$ and $\omega$ is an even $k$-form, then you need to choose an orientation on the tangent bundle of $N$ before you can integrate $\omega$ over $N$. If $\omega$ is an odd form, then you need to choose an orientation on $N$'s normal bundle in order to integrate (a fair number of arbitrary need to be made to specify such a thing, including, at least locally, a choice of normal bundle, but they turn out not to matter).

This treatment of orientation means that, for example, an odd $(n-1)$ form on an $n$-manifold can naturally represent a "flux": integrate over a hypersurface to determine the rate of transport through that hypersurface per unit time -- the transverse orientation tells you which direction of flow you consider to be "positive".

But the case that comes out most naturally is when you have an odd $n$-form on an $n$-manifold. In this case, when you integrate over an $n$-submanifold, the normal bundle is 0-dimensional, so there is a canonical choice of transverse orientation (between "$+$" and "$-$", just take "$+$"!), so no orientation data is really needed to integrate. This is analogous to the fact that no orientation data is needed to integrate an even 0-form over a 0-submanifold (i.e. to add up the values of a function at a set of points!).

The upshot is that a volume form or other nice measure is naturally an odd form, because the volume/measure of a submanifold certainly doesn't depend on any choice of orientation. Whereas when you try to treat volume as integration over an even form, you have to fix an orientation arbitrarily.