the badge is really the coat of arms of the city of Milano, where the company was founded and where its headquarters still are.
The emblem is split vertically into two halves: the cross on the left and the snake on the right. These were the symbols of the two ruling families of Medieval Milan which were adopted in the eleventh century. At that time, both families financed armies in the First Crusade. The local archbishop gave one of the armies a banner of a large serpent, said to be of Biblical origins, to carry into battle as a symbol of Divine protection. The other army, not to be outdone, adopted the crusader's red cross on a white field. The First Crusade was relatively successful and upon returning, a defeated Saracen was placed in the serpent's mouth as a symbol of victory.
In the fifteenth century, or thereabouts, the two families joined forces (and flags) to form the powerful Visconti Dukes. To signify royal consent of this merger, the Dukes of Austria (who were ruling Italy at the time) approved the placement of a crown on the serpent's head. Even after the power of the Visconti faded, the crowned snake and cross remained as the symbol of the city.