Once you defeat N Brio, you will unlock the ability to go through levels in 'N Verted' mode which flips the stage the opposite way as well as changing the artstyle to a different artstyle (there is a few different artstyles depending on the stage). All you have to do is finish any level in 'N Verted' mode in order to unlock this achievement.
This will add progress towards the achievement 'Backwards & Forwards'
Unlocking all of the gems in the 'N Verted' mode will add progress towards the achievement 'gnilB fo gniK'
Example of a N Verted stage:
Hope this guide helps, let me if know if I should add/change anything instead of downvoting :)