וכל עטיפה שאינה כעטיפת ישמעאלים,אינה עטיפה
There is a Machlokes Haposkim how to do the Atifah with the Tallis.
Some say to bring down the top of the Tallis above the eyes. Others say you may cover the eyes as well up to the mouth. Some Arabs in the Dessert cover their eyes to protect from the sand.
They can still see the ground they are walking on.
a)Those people who pull down the Tallis halfway over their head are not doing an Atifah (M.V. 8:sk4) (can’t see anything)
The Halacha requires to throw the Tzitzis to the back of the LEFT shoulder.
b) “Many Talmidei Chachamim mistakenly throw the Tzitzis also over their right shoulder. Doing so is בל תוסיף . There is no mention of it, not by the Pashtanim (Poskim) and not by the Mekubalim”.
(Piskei Teshuvos, Leket Hakemach )
c) Placing a folded Tallis on the shoulder, has nothing to do and is not part of Atifas Tallis.
Originally Shuls had no tables. The most convenient way of checking the Tzitzis, was by placing the Tallis on the shoulder. Checking the Tzitzis before making the Bracha is Halacha. Placing a Tallis on the shoulder is not.
Placing the Tallis on the shoulder and not checking the Tzitzis is Amei Ratzes. ( אמבצי)