If you use a short weapon, right swing is definitely a life saver, especially when approaching.
Using a short weapon with only 4 ath and medium-heavy armor, I need to rely on the range bonus from right swings. Every time I miss a left swing I think how a right swing would have done the job better and at least force defense from the opponent. Definitely requires more precise footwork and turning into swings than lefts, though.
I don't know if other people share the same sentiment as me, but I often confuse the beginning animations of left and overhead swings. Overheads are great even just to feint, imo.