Praetorians primarily functioned as a Secret Police during Imperial times, but it was born during the Republic era as an elite group made up of a specially chosen nine cohorts.

However, this special cohort transformed and became the Praetorian your are probably thinking of when Augustus came to power and made them his personal, imperial guards and gave them their own fortress in Rome, Castra Praetoria.

During later times, the Praetorians developed a special unit within their ranks referred to as the Speculatores which were covert, Roman, intelligence operatives embedded within Rome itself. It's during this time, that the Praetorians evolved from more a secret service into what we might think of as the CIA, on a far smaller scale.

It is rumored that the Speculatores had hands in several clandestine operations such as executing political enemies, although I can think of no quality stories to give you at this time.

The Praetorian guard eventually took this to the extreme, playing their part in the assassinations of Caligula, Commodus, Caracalla, Balbinus and Galba, as well as others.

It's really a fascinating story of the establishment of a truly iconic legion of hand-picked officers to guard Caesar, down to shadowy thugs who lurked about Rome killing and installing Emperors who appropriately bribed them, or refused to.

They were ultimate disbanded by Constantine (I think).

EDIT: I did not directly answer what type of relationship they had because the Praetorian's spanned centuries, and their role evolved and changed many times over. If you want to ask more specifically, I can re-examine.