Maybe im wrong but you can't change rotation on a single phase motor if it running off L1 and L2 at 240 volts

BTW, many do not quite understand the N.A. system of 120-0(E)-120.
When any two of the three points (120-0-120) are used, they are totally identical, just the voltage level changes, if you look at each or any pair on a 'scope it will confirm it and show simple/identical SP waveforms.

 

The two variations shown equate to what I suggested, It may work, or not.