If its a 3D game you simply could rotate the character by 180 degree (90/-90 for up/down) after pressing the corrisponding Directionkey. This would not mirror you animation, it would look different from side to side, but how different depends on the animation you have.
If its just a simple walking animation (left foot then right food forward) it would mirror it 99% (right food then left food forward), with the difference, that the right food would be on the other side, making it more realistic (only noticable if the Character looks different on one side).
And if you can give him wearables like armor that is shown on the Character and is not symmetrical (left shoulder with spikes, right without) it would show it only on the right side, while mirroring the Character would make him have Spikes on both sides, or on none.
Also mirroring only the animation would resuld in the charakter maybe raising the arm through the armor that is not there on the other side, so you would have always to consider this while adding new armor and animations, making it basically double the work (4 times with up and down).
I hope this Helps you.
Lexyth