  For each $\alpha \in \{0,1,-1 \}$, we count diagonally and antidiagonally
symmetric alternating sign matrices (DASASMs) of fixed odd order with a maximal
number of $\alpha$'s along the diagonal and the antidiagonal, as well as
DASASMs of fixed odd order with a minimal number of $0$'s along the diagonal
and the antidiagonal. In these enumerations, we encounter product formulas that
have previously appeared in plane partition or alternating sign matrix
counting, namely for the number of all alternating sign matrices, the number of
cyclically symmetric plane partitions in a given box, and the number of
vertically and horizontally symmetric ASMs. We also prove several refinements.
For instance, in the case of DASASMs with a maximal number of $-1$'s along the
diagonal and the antidiagonal, these considerations lead naturally to the
definition of alternating sign triangles. These are new objects that are
equinumerous with ASMs, and we are able to prove a two parameter refinement of
this fact, involving the number of $-1$'s and the inversion number on the ASM
side. To prove our results, we extend techniques to deal with triangular
six-vertex configurations that have recently successfully been applied to
settle Robbins' conjecture on the number of all DASASMs of odd order.
Importantly, we use a general solution of the reflection equation to prove the
symmetry of the partition function in the spectral parameters. In all of our
cases, we derive determinant or Pfaffian formulas for the partition functions,
which we then specialize in order to obtain the product formulas for the
various classes of extreme odd DASASMs under consideration.
