Base vs. Overlay
Rotating or flipping a base image stores a setting in the database that tells the program to make the adjustment to the display of the base image. The base image is not altered. However, adjusting an Overlay image creates a TIF copy of that image and all changes are made to that TIF and stored in an "Overlays" subfolder where the original Overlay image file is stored.
Page vs. Takeoff
When you rotate a Page, if it contains takeoff, that takeoff is rotated right along with the Page. However, if you want to only rotate takeoff you can do that without rotated the Page (the options are in the same menu). When you rotate just takeoff, if rotating it would cause it to no longer fit on the Page (Horizontally or vertically), you will be prevented from rotating the takeoff. See Related Articles for details on rotating takeoff.
Now, if you want to rotate the underlying Page without rotating the takeoff, you can open the Cover Sheet, duplicate the Page, navigate to the new Page, rotate it, and then copy/paste all the takeoff over. If it fits, it will be pasted over in its normal orientation, if the pasted takeoff will not fit (horizontally or vertically), you will be prevented from pasting it onto the new page.