In conventional coordinate systems (anything you solve a simple Newtonian mechanics problem with), up and down are + and - z. A vector pointing up and a vector pointing down are anti-parallel.
But in qm, we have up and down spinors making and orthonormal basis. These basis vectors are also called positive and negative z spin angular momentum. I understand the math for how spinors like (1,0) and (0,1) are orthogonal. I also see how they can be expressed as superpositions of x and y spinors, using complex numbers such that a two-component spinor can represent quantities in 3 dimensions. (this seems a little like what I have studied about symmetry groups like SU(1), so if that is relevant in the solution, I appreciate a discussion, but if it is unrelated, please do not bother correcting any huge mistakes in this sentence because I did not fully try to study it on my own yet).
My question is this: what is the intuition for saying that spin up and down are orthogonal?