There are various ways of constructing 5d SCFTs in String Theory; most famously, one can look at geometric engineering in M-theory or webs of 5-branes in type IIB. It is well understood how to translate from one setup to the other in the case where the Calabi-Yau geometry is toric. However, in the type IIB picture, brane manipulations such as Hanany-Witten transitions can lead us beyond the pure toric context; the combinatorial data enconding the system has been dubbed a Generalized Toric Polygon (GTP). In this talk, I will discuss recent progress understanding the geometry of GTPs. A key role is played by the mirror Calabi-Yau, where Hanany-Witten transitions take a very simple form. This allows us to make contact with a mathematical notion of "polytope mutation", and import part of the results in that literature to our physical setup; as an example, we find "mutation invariants" that can prove useful in the classification of 5d SCFTs. Time permitting, I'll also discuss some consequences for the BPS quivers of the 5d theories engineered by GTPs.