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We consider the alternating sign matrices of the odd order that have somekind of central symmetry. Namely, we deal with matrices invariant under thehalf-turn, quarter-turn and flips in both diagonals. In all these cases, thereare two natural structures in the centre of the matrix. For example, for thematrices invariant under the half-turn the central element is equal $\pm 1$. Itwas recently found that $A^+_{HT}(2m+1)/A^-_{HT}(2m+1)$=(m+1)/m. We conjecturethat similar very simple relations are valid in the two remaining cases.

1/N phenomenon for some symmetry classes of the odd alternating sign matrices