Flipping Layout in Altium
It's not clear to me whether this will preserve the orientation as you hope, but you can try this:


Shift-click all the objects to be moved (you can also shift-drag a
box around them).  Don't include any vias.
Shift-doubleclick one of the selected items.  The PCB Inspector dialog should pop up.  Properties which are common to all objects will appear here.
If every object selected is on the bottom layer, you should See "Bottom Layer" next to "Layer" (Otherwise you will see &lt;...>).  Click on this and change it to "Top Layer".