the resulting hosted cloud interactive object does not allow one to interactively rotate the contained graphic with the mouse as one would be able to do in a Notebook.
In contrast simply Cloud deploying the graph does permit this as in
CloudDeploy[Plot3D[Sin[x y+3],{x,0,3},{y,0,3}]]
Is there a clear reason for this difference in functionality? Or is it a bug or “coming feature”?
It can be rotated (but changes its appearance and size when doing so) some of the time. But after rotating it snaps back to its original orientation after a brief delay. Sometime then it completely stops working and one can only select the graphic.
We improved our Manipulate support some time ago, but you may or may not have tried your example recently. You can now interact with Graphics and Graphics3D within Manipulate.