To complete the 4 mirrored chapters is necessary not sufficient to get this achievement. Meaning that you won't get it at the end of mirrored chapter 4.
*** Spoiler - click to reveal ***You need to cross the mirrored bridge, and go back to your colored and torn down house. There you need to get the little white cube that moves.
Once you get that, you end the game, get the achievement, and go back to your black and white world where you will be able to select any chapter, level, mirrored or not.
There is a little spoil in the video, careful guys !