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A<sc>bstract</sc> We make progress in understanding the geometry associated to the Generalized Toric Polygons (GTPs) encoding the Physics of 5d Superconformal Field Theories (SCFTs), by exploiting the connection between Hanany-Witten transitions and the mathematical notion of polytope mutations. From this correspondence, it follows that the singular geometry associated to a GTP is identical to that obtained by regarding it as a standard toric diagram, but with some of its resolutions frozen in way that can be determined from the invariance of the so-called period under mutations. We propose the invariance of the period as a new criterion for distinguishing inequivalent brane webs, which allows us to resolve a puzzle posed in the literature. A second mutation invariant is the Hilbert Series of the geometry. We employ this invariant to perform quantitative checks of our ideas by computing the Hilbert Series of the BPS quivers associated to theories related by mutation. Lastly, we discuss the physical interpretation of a mathematical result ensuring the existence of a flat fibration over ℙ¹interpolating between geometries connected by mutation, which we identify with recently introduced deformations of the corresponding BPS quivers.