Is there a way to do this, using a concealed hinge? My guess would be that it would involve rotating the cabinet 90 degrees, onto it's side, but I wasn't sure if this could be done.
Yep, that's how you could do it. You probably want to change the layout of the tops and decks with reference to the sides since the sides will now actually be tops and decks and vice-versa.
In the cabinet editor double click the cabinet to select it
Click the Rotate Icon or right click then rotate item
rotate the cabinet 90 degrees on the Z axis
save cabinet as an assembly so that it will keep the rotation.