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:
$ cs2cs +proj=latlong +datum=WGS84 +to +proj=latlong +datum=WGS84 +pm=180dW
I get this:
0.0 0.0 180dE 0dN 0.000 1 0 181dE 0dN 0.000 179 0 359dE 0dN 0.000 -179 0 1dE 0dN 0.000
Which seems to be what you want.
I'm not sure whether this parameter will be honoured by all applications however. That page says:
Currently prime meridian declarations are only utilized by the pj_transform() API call, not the pj_inv() and pj_fwd() calls. Consequently the user utility cs2cs does honour prime meridians but the proj user utility ignores them.
So it'll depend on what you're doing with this CRS, and how the software you use goes about transforming points.