The halves are called Rive Gauche which means Left Bank and Rive Droite which means Right Bank. I learned about this "tradition" from a French student that was showing me around when I first arrived in Paris. While walking around the Left Bank, I noticed that many shops and venues had signs that said "Rive Gauche" beneath the store's name. I didn't really understand what it meant so I asked her about it and was instantly intrigued by her answer. She explained that for generations, each side of the river has been in constant competition with the other to be the 'best' side in terms of having the best food, shopping, monuments, neighborhoods and art.
Historically, Rive Gauche has been considered a more bohemian and free-spirited area filled with artists and intellectuals. This happens to be the side where I lived in Paris, so it will always have a special place in my heart. It is also well-known for being the home of some of France's most iconic monuments like the Eiffel Tower, Les Invalides (a military museum) and The Catacombs as well as museums like The Musee D'Orsay and The Rodin Museum. Its pride and joy are its trendy neighborhoods of Saint Germain Des Pres (where I lived and went to school) full of popular restaurants and shops and Le Quartier Latin (The Latin Quarter) best known for its lively atmosphere, student life and the prestigious university, La Sorbonne. While it does have good places to shop like Le Bonne Marche (one of the first department stores), I would say that the stores on the Rive Droite are at a whole different level.