Similar answer, a friend recently asked about twisted forms for non-orientable manifolds, what I found was pages 79-88 in Bott and Tu, "Differential forms in algebraic topology." I also think a twisted Stokes' Theorem is possible, as they present an entire twisted de Rham complex. Anyway, take a look:
http://books.google.com/books?id=S6Ve0KXyDj8C&pg=PA79&dq=bott+tu+twisted&cd=1#v=onepage&q&f=false
I googled "twisted Stokes theorem." My friend Dmitry asked originally based on some physics inquiries. It appears that these physics people give a pretty direct discussion, maybe it is enough. "Foundations of classical electrodynamics: charge, flux, and metric" By Friedrich W. Hehl, Yuri N. Obukhov, (2003) Birkhauser
