I think if you have 120 and 240 it must have a shared neutral unless there are 2 completely separate windings and a meter should show you that (ohm scale). My bet is you have 120-0-120 just like your house. Have you tried measuring between the hots of the two 5-15s? Is that is 240? Is there continuity between those neutrals? (more evidence that it is likely center tapped).
The open question is if the neutral is bonded.
