How do I: Fake a 360 spin with an Orthographic Camera?

Hello everyone, I was working on a 2d orthographic game when I came across a problem: how to fake the effect of the player "turning 360 degrees" in a 2d environment (ex. instead of moving a target box left/right on the screen, when the player moves left or right, the camera is moved left/right instead).

I first thought, "just extend the texture to the left and right, that way when the camera moves, it will make the player feel like they are turning around." But eventually the camera has to "snap back" or return to a starting position to make the illusion of turning around complete....

I'm just stuck on how to do that for now. For the npc objects, I figure a trigger box to teleport them would do the job, would the same work for a camera? Or maybe some sort of ray cast to trigger it?

Any ideas?