You can't understand it from the outside. You can't find a meaning if you were not born in Florence.
There is really no point in trying to understand a visceral, violent and historic tradition such as Calcio Storico Fiorentino (known as "historic football").
It is something that is bounded with the blood and the bricks of the city for centuries, that goes way beyond the touristic face of the city - made of cool gelaterias and leather shops - and goes deep down in the cruel and real soul of Dante's city. 

Despite the name it really does not look like the contemporary football. It takes inspiration from Roman legionaries' game and it actually recalls more a crossover between rugby, handball, and boxing. Matches last 50 minutes and are played on a field covered in sand, twice as long as it is wide (approximately 80x40 meters). A white line divides the field into two identical squares, and a goal net runs the width of each end. Each team has 27 players and no substitutions are allowed for injured or expelled players. The ultimate goal of the game is to score a Caccia throwing the ball in the net on the short side defended by the opponent team. In the middle, there is a real dirty battle without any rules, just the referees are allowed to stop a fight, but they do not succeed every time.

It is a tradition that goes from the 16th century into the city walls and the current tournament is the modern re-enactment of a particular single game played on February 17, 1530. At that time Florence was besieged by the Imperial troops of Charles V, the Florentines gave a show of nonchalance and cockiness organizing a match in Piazza Santa Croce. The modern tournament is a re-enactment of that match and is played every year since 1930 among the teams of four districts of the city: the Rossi di Santa Maria Novella, the Verdi di San Giovanni, the Bianchi di Santo Spirito and the Azzurri di Santa Croce.

The contemporary tournament is played every year in front of the Basilica of Santa Croce, where Niccolò Machiavelli, Michelangelo, and Galileo Galilei are buried. In an electric atmosphere, all the neighborhoods participate to the game supporting their gladiators to win the game.
No money is involved, nobody gets paid, the winning team is awarded a white cow. They risk their lives for the glory.
Le Coq Sportif invited us to this year semifinal between Verdi and Bianchi: as we said, we didn't understand much of what was going on, but this is what we have seen.

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