Brotherhood of the Hoodwink
Annotation and notes
In colloquial English, to hoodwink someone is to trick them into thinking what you want them to, not necessarily what they should.
The hoodwink is a device used to calm and pacify a bird of prey, covering its eyes and ears, perhaps lulling it to sleep in a false night, until the time comes to loose it on prey. This makes the bird docile and easy to handle. Well, for every falconer except Hodgesaargh, perhaps.
A similar blinding hood, also called the hoodwink, is used in initiation rituals in Freemasonry and other related mystical societies. This makes the initiate vulnerable and more susceptible to whatever ritual indignities are inflicted on them during the rites. (Shea and Wilson list a few variants on the theme of initiation). Sceptical commentators have also used it as a metaphor for the way members of secret societies allegedly allow themselves to be blindly led by their leaders, to serve ends about which they know nothing: in the metaphorical sense, it could be that the hoodwink never comes off. (In a Discworld context, think Elucidated Brethren of the Ebon Night).
The in-fighting between the orders appears to suggest a link to those competing secret and occult societies, as discussed in Robert Anton Wilson and Robert Shea's Illuminatus! trilogy of novels.
Shea and Wilson, in their masterpiece trilogy dealing with paranoia and the nature of conspiracy theory, draw a world-picture of competing secret societies seeking to wrest control of the world from each other, whilst remaining obscure to established governments and the general public. (Sounds just like the infighting between the orders of wizardry in Unseen University in the pre-Ridcully days, doesn't it...)
To those of us who have read and appreciated the Illuminatus! cycle, the thought occurs that Terry Pratchett cannot be unfamiliar with these books, as the Discworld novels are packed with sly allusions...