Isn't that just inherent in binary operations? I feel like you are saying the same thing as: What is 0 sometimes people write 1-1 or 3-3 or -10+10.
by the linearity of ^, B = a^b = 1 * a^b = x*(x^-1) a^b = xa ^ (x^-1)b
So the bivector is the "hyperbola" of all touching parallelograms just like 0 is the set of all oppositely directed arrows of equal magnitude.
But the parallelograms only become a bivector after you wedge them. not before.
