Thanks, my eyes are starting to cross from staring at computer screens for so long (work and home, I need some non digital hobbies), so I didn't really explain myself very well. I've made the unwrap to work in Maya, which is the 3D software I'm currently most familar with. I may even use the height map to do some displacement and model the world in 3D at some point. So the second image is the 3D unwrap.

However, I also want to present the map in 2D, so I was wondering if anyone knew of a good way to present a world map with the unfolded box layout I have above (the first image). The two parts I like best are the poles, so distorting it like how earth is often presented would be a waist :/ In 3D, I would call this layout a box sphere. Is there a term or technique like this in cartography to avoid distorion?