In 2D the only way to solve this is by having two different sets of animations for each side, no hidden tricks.
Granted, most games (including high profile ones) don't even bother with it and just mirror the images.
The most common example of this is 2D fighting games, at all levels of development, from the tiniest of studios to big budget AAA games, the graphics are just flipped and nobody seems to be bothered with it. Some of the newer fighting games that could easily do it because they are in 3D, still choose to flip the assets because it is more convenient for the players to see the same graphics no matter what side the character is facing.
If you still want to do it, you will need to produce more graphics.