Solution:

Let us color the cards of the deck blue and red, in alternating small groups of three and four respectively. The three at the top are blue, the next four are red, the three after that are blue, and so on. We observe that, when the maga Magò flips a packet, this alternating group coloring is preserved. If we focus our attention on the blue cards only, we see that there are eight groups of three cards, and at each move the topmost group is placed at the bottom of the sequence flipped over. Eight moves are therefore needed to flip over all the groups, and another eight to put them back in their initial orientation. As a result, the blue cards return to being all facing downward every 16 moves. The same happens for the red cards, which form seven groups of four cards, so they return to their initial state every 14 moves. All the cards will then be facing down again for the first time after $112=\operatorname{mcm}(16,14)$ moves.