  We discuss a family of operators which commute or anti-commute with the
twisted transfer matrix of the six-vertex model at $q$ being roots of unity:
$q^{2N}=1$. The operators commute with the Hamiltonian of the XXZ spin chain
under the twisted boundary conditions, and they are valid also for the
inhomogeneous case. For the case of the anti-periodic boundary conditions, we
show explicitly that the operators generate the $sl_2$ loop algebra in the
sector of the total spin operator $S^Z \equiv N/2 ({\rm mod} N)$. The
infinite-dimensional symmetry leads to exponentially-large spectral
degeneracies, as shown for the periodic boundary conditions \cite{DFM}.
Furthermore, we derive explicitly the $sl_2$ loop algebra symmetry for the
periodic XXZ spin chain with an odd number of sites in the sector $S^Z \equiv
N/2 ({\rm mod} N)$ when $q$ is a primitive $N$ th root of unity with $N$ odd.
Interestingly, in the case of N=3, various conjectures of combinatorial
formulas for the XXZ spin chain with odd-sites have been given
\cite{Odd,Razumov,BGN,GBNM}. We also note a connection to the spectral
degeneracies of the eight-vertex model.
