Given a surface $S \subset \mathbb{R}^3$ and its parameterization $\phi:(u \in[a,b],v \in[c,d]) \rightarrow \mathbb{R}^3$, we can have the normal vector as $\vec{n} :=\phi_u \times \phi_v$, where these represent derivative vectors of $\phi$ wrt. the indexing variable, as defining the positive orientation of $S$.
However, if we instead use $ \vec{n} := \phi_v \times \phi_u = -(\phi_u \times \phi_v)$, we get normal vector in opposite direction, i.e. giving a orientation-reversing parameterization of the surface.
That is, just by changing order of cross product, we can get orientation-reversing or preserving parameterization of the surface. But then, it doesn't make to sense talk about whether a parameterization is orientation-preserving or reversing? How do we reconcile this?