Another seven people are under investigation in connection with the gang, which also operated in rental car parks and at the main train station Termini, police said.
The suspects are accused of nabbing property worth more than €50,000 in a matter of hours, according to police, who released surveillance camera footage of the gang at work.
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The video shows several men, posing as passengers and carrying suitcases of their own, striking opportunistically at check-in desks or in the car park.
In one incident, a man can be seen opening a car door to snatch valuables from inside the vehicle as two travellers load their cases into the boot. In another, one man distracts a man waiting in line as an accomplice lifts his bag from a luggage trolley.
The gang has been dubbed the "commuter" thieves, because the suspects are residents of France and Spain but travelled back and forth to Rome.
Police said they were skilled at blending in with other travellers and carried changes of clothes with them to be able to swap outfit rapidly.
They face charges of aggravated theft.
Meanwhile police were able to recover some of the stolen property and return it to its rightful owners, they said.