  We investigate symmetry properties of the Bethe ansatz wave functions for the
Heisenberg $XXZ$ spin chain. The $XXZ$ Hamiltonian commutes simultaneously with
the shift operator $T$ and the lattice inversion operator $V$ in the space of
$\Omega=\pm 1$ with $\Omega$ the eigenvalue of $T$. We show that the Bethe
ansatz solutions with normalizable wave functions cannot be the eigenstates of
$T$ and $V$ with quantum number $(\Omega,\Upsilon)=(\pm 1,\mp 1)$ where
$\Upsilon$ is the eigenvalue of $V$. Therefore the Bethe ansatz wave functions
should be singular for nondegenerate eigenstates of the Hamiltonian with
quantum number $(\Omega,\Upsilon)=(\pm 1,\mp 1)$. It is also shown that such
states exist in any nontrivial down-spin number sector and that the number of
them diverges exponentially with the chain length.
