Ancient and Truly Original Sages of the Unbroken Circle
One of the eight orders of wizardry, aka The Illuminated Brothers of the Unbroken Circle.
Not to be confused with the Illuminated Brethren of Ee, who were a more part-time and occasional secret brotherhood who were quite probably the "Territorial Army" equivalent - meeting one evening a week and some weekends (as their day jobs allowed) provided the Fretwork Club didn't want the room. The Head of the Order at the time of The Light Fantastic was Greyhald Spold. Another luminary of the Order, in the time of The Colour of Magic, was Rhunlet Vard.
Again this appears to link to a name of one of the many competing secret and occult societies, which appear 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 Unseen University in the pre-Ridcully days, doesn't it...)
The Cult of the Unbroken Circle are a secret society purporting to have an unbroken history and continual existence all the way back to Atlantis... (compare this to the Freemason's claim to represent an unbroken continuity of mysticism that goes back to Hiram, master mason and architect of Solomon's Temple...) On one level Shea and Wilson are spoofing the Masons. On another, the suggestion is made in the trilogy that the Atlantis claim might actually be correct...
To those of us who have read and appreciated the Illuminatus! cycle, the thought occurs that Terry Pratchett cannot be unfamilar with these books, as the Discworld novels are packed with sly allusions...
Their alternative name "The Illuminated Brothers of the Unbroken Circle" sounds like an absolute giveaway in this context!