I think that he wants a generalized equirectangular projection to allow spinning the globe under the projection (if that makes any sense). The Cassini projection is the transverse of the Plate Carree projection, but I think that he wants a generalized one. My brain won't spit out the name of that one right now, though, so I'm not much use on that front.

While he certainly can convert his equirectangular input to something like Orthographic, move the center of projection to move the poles, and then unproject back to Equirectangular, there will likely be some unpleasant distortions in the process that will require high resolution images to reduce the error. He should be able to do all of that using something like gdalwarp, but that seems hard...