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