(How) can I determine the characteristics of an unidentified SMPS transformer?
There does appear to be a published UL document that contains all the build and test info for a transformer itemized as "E150436 C". This is just a snippet: -



The UL document can be found here and maybe this transformer has been used in a couple of products. It's towards the end of the document that you get the coil stack diagram, coil turns and pin-outs.

Just check to see that it looks the same before doing anything risky.