  In most of today's exactly solved classes of polyominoes, either all members
are convex (in some way), or all members are directed, or both. If the class is
neither convex nor directed, the exact solution uses to be elusive. This paper
is focused on polyominoes with hexagonal cells. Concretely, we deal with
polyominoes whose columns can have either one or two connected components.
Those polyominoes (unlike the well-explored column-convex polyominoes) cannot
be exactly enumerated by any of the now existing methods. It is therefore
appropriate to introduce additional restrictions, thus obtaining solvable
subclasses. In our recent paper, published in this same journal, the
restrictions just mentioned were semidirectedness and an upper bound on the
size of the gap within a column. In this paper, the semidirectedness
requirement is made looser. The result is that now the exactly solved
subclasses are larger and have greater growth constants. These new polyomino
families also have the advantage of being invariant under the reflection about
the vertical axis.
