Three-axis obstruction to unitary qubit orthogonalization

No fixed unitary operator maps both eigenstate rays of each of the three Pauli matrices to their orthogonal partners. Such a map would give $U^{†}XU=-X$, $U^{†}YU=-Y$ and $U^{†}ZU=-Z$. But unitary conjugation preserves multiplication. The relation $XY=iZ$ would imply both $U^{†}(XY)U=iZ$ and $U^{†}(iZ)U=-iZ$, a contradiction.

Two axes alone do not suffice: $U=Y$ flips the X- and Z-basis labels while preserving the Y-basis labels. Thus six specified input rays already obstruct a unitary implementation, without requiring tests on every point of the Bloch sphere.