  The nineteen-vertex model on a periodic lattice with an anti-diagonal twist
is investigated. Its inhomogeneous transfer matrix is shown to have a simple
eigenvalue, with the corresponding eigenstate displaying intriguing
combinatorial features. Similar results were previously found for the same
model with a diagonal twist. The eigenstate for the anti-diagonal twist is
explicitly constructed using the quantum separation of variables technique. A
number of sum rules and special components are computed and expressed in terms
of Kuperberg's determinants for partition functions of the inhomogeneous
six-vertex model. The computations of some components of the special eigenstate
for the diagonal twist are also presented. In the homogeneous limit, the
special eigenstates become eigenvectors of the Hamiltonians of the integrable
spin-one XXZ chain with twisted boundary conditions. Their sum rules and
special components for both twists are expressed in terms of generating
functions arising in the weighted enumeration of various symmetry classes of
alternating sign matrices (ASMs). These include half-turn symmetric ASMs,
quarter-turn symmetric ASMs, vertically symmetric ASMs, vertically and
horizontally perverse ASMs and double U-turn ASMs. As side results, new
determinant and pfaffian formulas for the weighted enumeration of various
symmetry classes of alternating sign matrices are obtained.
