Mr. Becker is correct. The 2 hot legs of typical USA household 240VAC service are each at 120VAC potential (to ground), but their wave forms are 180 degrees out of phase (offset from each other), so the simplistic result is that at any particular moment you measure from the 'peak' of one to the 'valley' of the other - - and get 240V. This 240V is what a meter should 'see', as well as your 240V motor, etc.
If you measure 120V on L1<>G and L2<>G, but get (zero) 0V from L1<>L2, then the receptacle is incorrectly wired for use with anything. The only way anything will function on this is if it uses the Gnd lead as a conductor. Generally speaking, this would also imply there is a problem with the wiring in the 'motor' allowing current to flow to the ground, but this opens a whole new rabbit hole of speculation and conjecture.
Photos of meter on a 240V AC circuit, and what it should read on a properly wired 240VAC plug, like the OP's follow:
This from left leg to ground--
Attachment 353811
This from right leg to ground--
Attachment 353812
This from left to right leg--
Attachment 353813
For some who have responded, please ask yourself why it's called "240V" if you don't measure 240V somewhere.