Not certain about the 60D, but I had an issue many years back with a Powershot G2 screen due to a connector related issue. In my instance the problem would clearly go away when I pressed on the area where the hinge was located, which is also the connector area. So you might try pressing around the hinge on your 60D when the image is acting up.
In my instance, with a small set of jewelers screwdrivers I carefully removed the screws around the screen frame and then removed the frame. In my instance the connector was indeed partially disconnected. I setaed the connector fully and put some soft foam between the connector and the frame, then reinstalled everything. I never had the issue again.
If you try something like this, have some tweazers handy and place each screw as you remove it on a strip of scotch tape. Otherwise, should you drop a screw, it is near impossble to find it! I also put a very small amount of clear nail poilsh on each screw when I reassembled the camera.
EDIT; added info...Some further research on the G2 indicates that "23cat" posted the response immediately below about the G2 problem they faced (screen turned off when in body). (http://www.fixya.com …rns_off_when_back_in_body
)
"We had a similar problem. If you remove the cover to the place where the LCD hinges, you will find two little wires broken off. These attach to a black touch sensor resting in the lcd screen. It comes out easily and without issue. WIth the two wires torn, the camera does not recognize that the LCD display has been turned around. It turns the screen off because it believes you have the LCD screen pointed inward towards the body of the camera. A small extension for the wires and a soldering iron solved that set of problems."
