All dogs in the cities are honourary members of the Order, and the alpha dog of the city is given the title of Lord Chancellor, and the human head of the Order is given the title of Chancellor's Man. Aside from the usual responsibilities you'd expect of the head of the judiciary and the supreme commander of the city guard, and the Loyal Order's representative in the Mayor's Privy Council, he also has ceremonial duties involving feeding the Lord Chancellor and taking him for walks. In the old days the Lord Chancellor was often a fierce wolfhound or some other such dog; these days it tends to be a very big poodle of regal bearing.
The laws of Horizon are broadly similar to the sort of laws you'd expect in an oligarchic city state at the nexus of reality: Don't kill people. Don't steal from people. Don't incite revolution. That sort of thing. A few interesting quirks:
(NOTE: The recent case of Steam Union and Clockwork Guild vs. Grey Order concluded with a ruling in favour of the Grey Order: since the Order only shared trade secrets amongst its members, it couldn't have been said to revealed them to the public.)
The human as well as the canine faces of the law are all affiliated with the Loyal Order in some way. It employs and pays the Horizon Guard - known colloquially as the Watchdogs. Judges are high-level members of the Order; advocates must pay the Order a stiff tax in order to represent people within courts. The Order also runs the Doghouse, Horizon's vast prison; the labour of the prisoners has proven an excellent way to gain extra funds.
There's a rotten secret at the heart of the Doghouse, by the way: lately, a group of embittered ex-military/guard sorts disgruntled with the wildly corrupt Watchdogs of today and inspired partially by nostalgia for the Imperial days when everyone lived in fear, partially by the ravings of Old Cracky, a Star-inspired wizard who was sentenced to life imprisonment a hundred years ago for trying to overthrow the Emperor and institute Cracky's vision of the Government of the Stars. Cracky has taken to, with the aid of certain guards whose aid he has enlisted, sucking the minds out of prisoners who are about to be released and filling them with starlight so that they will go out and impose a Higher Law on the mean streets of Horizon. He's not got a production line going here, mind - not all the prison wardens are in his cult, so he only gets to subvert one or two prisoners a month - but things are going to start getting interesting... oh, around the time the campaign starts.
The Loyal Order also makes sure people don't meddle with the dogs. Killing or hurting a dog is an abhorrent and taboo crime within Horizon; nonetheless, there are some strange people out there. Often the dogs do a good job of sorting out this sort of thing themselves: a pooch who witnesses a maniac murdering a chihuahua will pee on the appropriate lampposts and soon all the dogs in the city will be hunting the villain down. More careful folk can pose a problem; the Loyal Order's Guardsmen of Honour ("the MANHUNTERS") work diligently to root these people out.
There are not that many subdepartments within the Loyal Order: the custodians of the law. Lately, however, a number of retired and fired guardsmen have formed various private detective agencies, who have proven popular amongst those who aren't quite satisfied with the service they've been getting from the Guard. High-ups in the Hounds tend not to mind this - anything which takes the burden off their shoulders is a good thing by their book - but rank-and-file Watchdogs don't take kindly to PIs poking around their cases.
It has been known for the Loyal Order to hire detective agencies on a long-term basis to perform some task; for example, Willie Frank's Investigative Bureau has been engaged to keep the semi-constant battles between the mobs churning on at an acceptable level; not quite violent enough to disrupt the day-to-day life of Horizon, but bloody enough to keep going under their own steam. The intent is to make sure that the mobs expend most of their resources fighting one another instead of, say, banding together to overthrow the Mayoralty and carve up the city between themselves.