How about you get square piece of wood, cut a holw in the centre for the screen to poke through nice and snug, and then screw those 4 front mounting holes to the wood. Now mount the wood to the cabinet.
Cut two handles in the wood, one top and one bottom. Now if you decide you want to play a vertical game you just release the wood, grab the handles, pull the screen out and rotate it and plop it back in the cabinet.
Simple but effective.