Left or right? Does the direction of necktie stripes hold any significance? 

Fashion folklore holds that the meaning of stripe direction of the has it's roots in military history. 

When soldiers have their rifles slung over their shoulders, the strap crosses their chests diagonally. In order to avoid a crisscross of the strap with the tie stripes, the tie stripes were worn in the same direction as the strap. 

Therefore, the American soldier, who carries his weapon slung over his right shoulder wears a striped tie that point to the right shoulder. This is called the "American Rep Stripe" tie. 

Conversely, the European soldiers wear their weapons slung over their left shoulders, and their striped ties run pointing to their left shoulder. And that is called the "European Rep Stripe" tie.