 Separate orbit polynomials can miss mixed invariants

Let $L=K(\sqrt{a_1},\sqrt{a_2})$ with independent square classes and characteristic different from two, and let $G$ be generated by the automorphism negating both square roots. The two orbit polynomials are $t^2-a_1$ and $t^2-a_2$, so their coefficients generate only $K$, whereas