Not all have left and right. Some styles do, many don't. The tennis ones definitely do, which I do not play tennis, but love how they feel on. I check the materials and get the ones that are higher in cotton, as I found I like them more than the ones higher in polyester (usually, the light weight "performance" socks.)
It feels like they offer more support, in the bee hive type band around the arch. I do know I have not had the plantar fasciitis act up since I have been wearing them the last 2-3 years.