I think maybe there's some confusion here regarding the definitions of symmetry and antisymmetry, because they're very closely related. Symmetry says "whenever

I think maybe there's some confusion here regarding the definitions of symmetry and antisymmetry, because they're very closely related. Symmetry says "whenever A, then B" and antisymmetry says "whenever A, then not B". I.e, given a piece of information A, symmetry and antisymmetry tell you to draw opposite conclusions, which is why one is "anti" the other. The only time a relation can be both symmetric and antisymmetric is when the antecedent in their definitions is never true, which is the trivial case.