I have a 220 V -> 110 V AC "wall wart" shaped step-down transformer with a plastic body. The input has an ungrounded three-prong plug (the ground is plastic) and the output is a polarized socket. The lowest resistance path between an input and output is the hot input being connected "neutral" output (fat side of a polarized US-style socket), with 0.2Ω resistance. My analysis is that the transformer's inputs are backwards. Is it worth plugging in the transformer backwards if it requires cutting and glue and makes a less secure physical connection afterwards?
Info about the application that may not be relevant:
I am using it to power a fan that has a fully plastic body. The fan has an internal DC adapter and is the internals run on 24 V DC. Is there anything unsafe about this?