As an Italian, I feel like this post may be misleading to people living in countries like the USA, even if I understand what you meant, and it is an interesting topic.

I think it is important to notice that in Italy police is not part of the military, they are civilians. They are under the authority of the Ministry of Interior. There no ranks, because rank is a military word, at least in Italy. There qualifications, which are kind of similar. They are very different from let's say the American police force; while Italian Police is an armed force, for the vast majority they deal with civilian stuff within the Italian national border. Hence why their link with sport is not as strange as it may seem.

Carabinieri are indeed a military force, under the authority of the Ministy of Defence. They can go fighting wars, and it's not rare for them to work abroad. There are ranks, and so on.