Summary: For two-way contingency tables, S. Tomizawa [J. Jpn. Stat. Soc. 15, 17–23 (1985; Zbl 0564.62047)] gave a theorem that the double symmetry (DS) model holds if and only if both the quasi DS and the marginal DS models hold. The present paper proposes, for multi-way tables, the DS, some quasi DS and marginal DS models, and extends Tomizawa’s theorem into multi-way tables. It also shows that for multi-way tables the likelihood ratio statistic for testing goodness of fit of the DS model is asymptotically equivalent to the sum of those for testing the quasi DS model with some order and the marginal DS with the corresponding order. An example is given.