Can't seem to find anything on it, or rather, I don't quite know what to search.
As seen here on an American uniform, and here on French uniforms, it looks like a blanket, tied across their chest. I've also seen it on Imperial Japanese uniforms.
I'm assuming it's the soldier's personal items rolled into a kit, and the "blanket" is just that. But is there another practical purpose of it? Surely it would not stop projectiles.
Is there a proper term to call it?
Those are greatcoats or cavalry cloaks that have been rolled up before a battle in order to provide some protection against sabre cuts and bullets. Cuirassiers during the Napoleonic era would also sometimes wear it around the pommel of the saddle to protect the stomach and groin. For a cavalryman, the rolled cloak would also keep the pistol holsters clear of obstructions, however there was also the danger that during a charge an enemy could grab the coat and unhorse the rider.
On the march, infantry would usually carry their rolled up coats on the tops of their packs (which the Zouave in the picture appears to be lacking, replaced by a haversack on his left hip) and cavalry would carry theirs on their portmanteaux (travelling cases) behind their backs on their saddles. Some soldiers also found it more comfortable to wear the coat rolled up over the shoulder when marching.
As an aside, during the Napoleonic era a sash usually refers to the brightly coloured material worn like a belt around the waist by officers and Hussars rather than the ceremonial over-the-shoulder type usually referred to as a sash today.
Sources:
Swords Around a Throne - John Elting
Napoleon's Infantry Handbook - Terry Crowdey
Light Cavalry Outposts - Antoine de Brack
