  We calculate the field of rational local unitary invariants for mixed states
of two qubits, by employing methods from algebraic geometry. We prove that this
field is rational (i.e. purely transcendental), and that it is generated by
nine algebraically independent polynomial invariants. We do so by constructing
a relative section, in the sense of invariant theory, whose Weyl group is a
finite abelian group. From this construction, we are able to give explicit
expressions for the generating invariants in terms of the Bloch matrix
representation of mixed states of two qubits. We also prove similar rationality
statements for the local unitary invariants of symmetric mixed states of two
qubits. Our results apply to both complex-valued and real-valued invariants.
