(looking from lock side on right handed rifle )
I'd guess the other way (but only an opinion) if it has to be on the top sign it so it can be read when the rifle is held, the way it will be held in use (right hand on wrist, left hand on forearm) the letters presenting themselves from left to right. Do not force some one to look at upside down or back wards writing all day while they sit on a stump and try to admire the gun on their lap.
This will be opposite to how you would want it presented for a rifle on a stand or on a wall or on an internet show and tell pic. People who are looking at a rifle they are not holding can walk away the hunter is stuck with it.
Just another opinion but I find it very unique to see a signed barrel that does not detract from the overall gun. A touch mark or tactful stroke is a bit classier then a graffiti style tag on a perfectly struck, draw filed and fitted rifle barrel! Odd practice, Imagine if the guys at the mine, the steel mill, the rolling mill, the boring and rifling shop and then the assembler all signed the barrel LOL.