Context: I am trying to determine whether or not there exist generalizations for the notion of orientation, and if they do exist what are they. In order to determine whether or not these generalizations exist, I need to view orientations as an equivalence relation and then abstractly consider additional equivalence relations that can be defined.
Consider the set of all possible bivectors $\mathfrak{B}$ in $\mathbb{R}^3$. Then there are three possible equivalence relations.
Equipollence: The equivalence relation $(\mathfrak{B}, \sim)$ such that a ~ b whenever a has the same normal vector as b.
Orientation: The equivalence relation $(\mathfrak{B}, \sim)$ such that a ~ b whenever the boundary of a rotates in the same way as b.
Size: The equivalence relation $(\mathfrak{B}, \sim)$ such that a ~ b whenever Area(a) = Area(b).
Now is there anything else that singles out these three equivalence relations as special? There are many other equivalence relations we can put on $\mathfrak{B}$, like equivalence depending on whether one of the edges is the same. But they all seem stupid. These three equivalence relations seem special compared to all of the others. Why?
$\mathbb{R}^3$ is special in the sense that the dimension of the bivector space is the same as the original space (which is usually not the case).
Under the usual identification of bivectors with vectors via the usual cross product we get that equipollence is the same es representatives being in the same linear subspace and for size we get that representatives have the same magnitude.
I'm not sure what you mean by 'boundary rotates in the same way'. Orientation is a good equivalence class for n-vectors in n-dimensional space.
Similarly 'compare one of the "edges" ' is not really an equivalence relation as $ a \wedge b = - b \wedge a $ so we'd have a problem with reflexivity
The other two equivalence relations are quite special as you can see they are invariant under action of $ SO(3) $ (the usual rotations) on your vector space. Any symmetric partition of $ \mathbb{R} $ can be extended into an equivalence relation of a similar kind: e.g. all bivectors with size < 1 are equivalent and all others are equivalent only if they are proportional. I guess there is some additional level of 'specialness' in the choice of equivalence classes on $ \mathbb{R} $ in the examples you've chosen but I'll leave it at that...