  We obtain asymptotic formulas for the partition function of the six-vertex
model with domain wall boundary conditions and half-turn symmetry in each of
the phase regions. The proof is based on the Izergin--Korepin--Kuperberg
determinantal formula for the partition function, its reduction to orthogonal
polynomials, and on an asymptotic analysis of the orthogonal polynomials under
consideration in the framework of the Riemann--Hilbert approach.
