4D Duoprisms based on nonconvex polygons

Question

A duoprism is a polytope that can be expressed as the Cartesian product of two polytopes (each of dimension $\ge 2$). Four-dimensional duoprisms in particular have been studied: $$P \times Q = \{ (x_1,x_2,x_3,x_4) \;\mid\; (x_1,x_2) \in P,\; (x_3,x_4) \in Q \} \;,$$ where $P$ and $Q$ are $2$-dimensional polygons.

All the literature I've seen assumes $P$ and $Q$ are convex polygons. For example, here is a net for the product of a square and a regular hexagon:

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                ^{(Image from Wikipedia page.)}

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Q. Does anyone know of any investigations or uses of duoprisms in which either or both of $P$ and $Q$ are nonconvex polygons?

Answer 1

           There is some information about the product of two pentagrams here

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Answer 2

The great duoantiprism can be constructed using a compound of two pentagonal-pentagrammic duoprisms by inserting additional edges, or by alternating a decagonal-decagrammic duoprism.