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Your conclusion that I think a pseudovector is actually a function of vectors equipped with preallocated arguments and every transformation on a vector has a counterpart on a pseudovector like $\tilde P$. is, I think, essentially correct; the reason parity is confusing is that we tend to drop the tilde. More precisely, whenever parity considerations ...
This all links back to the fact that any time you speak about a transformation, you require all the relevant physics to transform along. E.g. velocity would not transform the way it transforms without requiring the trajectory $x^i(t)$ it is tangent to to transform also. In the sense of the notation you use we have $v \equiv <x^i(t),\frac{d}{dt}>$ and ...