 Collective-unitary covariance of the two-qubit singlet

For $|s⟩=(|01⟩-|10⟩)/\sqrt{2}$ and any two-by-two unitary matrix, expand the two columns of $U$. The equal-index coefficients cancel, and the coefficient of $|01⟩-|10⟩$ is the determinant. Its modulus is one, so a common change of qubit basis preserves the spin singlet state up to a global phase. The density projector is exactly invariant.