The player is led to assume they're viewing everything through Junpei's perspective, but only the top screen truly represents him. Everything you saw on the bottom screen was actually past Akane's perspective, because she's been psychically guiding Junpei to help him solve every puzzle. I'm glossing over a complicated plot about seeing through time, you'll have to play the game for it to make sense.

But past Akane couldn't solve the final puzzle in her own time, and would've have died. She's revealed to be the one who orchestrated all of the events of the game just so that future Junpei would develop the same psychic time powers that would enable him to guide her past self through that puzzle, closing the loop and saving her life.

So for the final puzzle, the player must turn the DS upside-down so that the touch screen is now on top, representing Junpei solving the puzzle himself in order to guide Akane.

It's a bit of a tragedy that this detail had to be sacrificed in the single-screen HD remake, because it's so damn clever.