The tensor product maps the vector space into which is a different vector space, so the notation is misleading; it should be (upper or lower indices depending on co- or contra variance, if such notation is relevant at all.
Whether the original vector space remains (the new space is "created" in addition) or whether the original vectors space is "destroyed" is a question of notation and context (one can destroy the "cross product" by simply changing the order or multiplication and adding) in a casual presentation.
However, arithmetic processes between scalars in a given vector space do not affect the sign of the vector product. It is the vector itself that changes sign (a "mirror" rotation in Lorentz space). (That is why the strict definition of cross product is really:
Changing the sign of scalars is a different process than changing the parity of the vectors.
The "outer" product is distinguished for the cross product (which relates to n=2 and the "right hand" rule) for this reason (and a number of others, including parity).
The problem the cross product raises is apparent in the problem of gimbel locking, which is why quaternions are used in action adventure games requiring rotation, among other applications.