The Bat-signal is problematic for Gordon. I think the relationship that Gordon and Batman have in the Legends of the Dark Knight stories works much better.
Anyway, I've always seen Gordon's relationship with Batman as something similar to the relationship some police officers have with private investigators. They collaborate on certain things and share information when it suits each other's needs, but by no means is the private investigator an arm of the police, officially or unofficially. It's not a partnership but more an unofficial relationship of convenience.
