When talking about physics (on $\mathbb{R}^3$), one tends to call pseudovectors $+$-parity objects, since they transform into themselves under the parity transformation $P: (x,y,z) \mapsto (-x,-y,-z)$. Similarly a pseudoscalar is a $-$-parity object. One can clearly see how this corresponds to the degree of a form in this space as pseudoscalars are $3$-forms, and pseudovectors are $2$-forms.
I've seen it written elsewhere that on an $n$-dimensional manifold, $n$-forms are always pseudoscalars, and $(n-1)$-forms are always pseudovectors, but this seems wrong since for even $n$ pseudoscalars transform like scalars and pseudovectors transform like vectors.
I've also heard from senior academics that twistedness (a.k.a. screw-sense, odd/even forms) is related to pseudo-quantities, but this doesn't make sense to me either since it was my understanding that twistedness relates the form's orientation to the manifold's.
What (if there is) is the generalisation of pseudo-quantities to general manifolds?