I thought I would post this question here rather than in the Classical Physics formum because I expect the GR experts might be better able to answer this. I'm trying to get a phyisical/intuitive/geometrical explanation for why the charge-density/current 3-form (sometimes called a 4-vector and often referred to as J) is a twisted form. By "twisted form" I mean a differential form that can be defined on a nonorientable manifold. It's clear to me why a 3-form is appropriate for J, but I can't seem to fathom why a twisted form is needed. I think a key part of my question is that I don't really understand (aside from some non-intuitive mathematical statements) the important differences between a conventional differential form and a twisted differential form. Any insight, even if incomplete, will be greatly appreciated.