Let be a complete discrete valuation ring of mixed characteristic with perfect residue field. From the semi-stable conjecture () and the theory of slopes, we obtain isomorphisms between the maximal unramified quotients of certain Tate twists of -adic étale cohomology groups and the cohomology groups of logarithmic Hodge-Witt sheaves for a proper semi-stable scheme over . The object of this paper is to show that these isomorphisms are compatible with the symbol maps to the -adic vanishing cycles and the logarithmic Hodge-Witt sheaves, and that they are compatible with the integral structures under certain restrictions. We also treats an open case and a proof of in such a case is given for that purpose. The results are used in the work of U. Jannsen and S. Saito [Doc. Math., J. DMV Extra Vol., 479--538 (2003; Zbl 1092.14503)].