After finishing the main quest in the Shivering Isles you fee Jyggalag with the Prince himself remarking, "Now, though, you have ended the cycle. You now hold the mantle of madness, and Jyggalag is free to roam the voids of Oblivion once more. I will take my leave, and you will remain here, mortal. Mortal...? King? God? It seems uncertain. This Realm is yours. Perhaps you will grow to your station. Fare thee well, Sheogorath, Prince of Madness.". That was my While reading posts on various forums I found a link to an article written for the Imperial Library which made the argument that the two were inseparable, however Jyggalag was initially solely order before the curse placed upon him by the other princes so why would the mantling of one necessitate the mantling of the other?
link to article
http://www.imperial-library.info/content/arden-sul-1