[seqfan] Re: Counting polyominoes with peculiar symmetry

[Richard J. Mathar]

Such "diagonal" spanning skeletons (one or two for each polyomino) are square 
polysticks, A019988 .  Some square polysticks cannot represent a "diagonal"
spanning skeleton of a polyomino --- so they "break" that kind of duality.
For example all square polysticks which are just a sequence of straight line 
segments all in the same direction cannot be "diagonal" skeletons
because these cells would not be edge-connected.
In https://commons.wikimedia.org/wiki/Category:Polyominoes
there are illustrations of polyominoes split by symmetry classes,
see e.g. https://commons.wikimedia.org/wiki/Category:Nonominoes .