What is the between normal and local space? So I get the difference between global space (or world space, whatever you want to call it) and local space-- global space looks at locations in a 3D Cartesian Graph that has a fixed orientation/location which is independent of the objects within it while local space bases its 3D graph's orientation/location off of the object that that particular space belongs to. Then normal space is dragged into the mix and it gets confusing: I've tried to spot a difference between local and normal space by using the transformation orientation and, as best I can tell, normal space and local space operate with the same orientation/location.
Probably your confusion derives by using a cube as example, which is an ultra regular object. Try Suzanne instead: the difference between local and normal is evident. The normal is calculated by the average of all normals of selected faces.