I need the coordinate system reference or a projection definition that uses geographic (i.e. lat/lon) coordinates but instead of east/west bounds being defined as 180 to -180, I need them to be defined as 0 to 360. In other words Greenwich would be at 180 degrees longitude.
Looking here, you can use the +pm parameter to specify your own prime meridian relative to Greenwich, so using this command:
I get this:
Which seems to be what you want.
I'm not sure whether this parameter will be honoured by all applications however. That page says:
So it'll depend on what you're doing with this CRS, and how the software you use goes about transforming points.