Is there any free service to which I could send a UDP packet and it would respond telling me what is my "real" UDP port? (my application is behind a NAT)
This kind of service could allow me to make a p2p coordination server with normal php hosting.
Such service is called STUN. If you query a search engine for public stun server you should get a bunch of results. There is an RFC on STUN servers, but I would be surprised if any public server would conform to that.