Developing the concept further, there are two ideas here. The first, is that the pairs represent mental dispositions:
White <--Attitude--> Black
Grey <--Liveliness--> Green
Brown <--Alacrity--> Yellow
Blue <--Dedication--> Red
Clear <--Method--> Violet
This would make the various colors opposed to each-other. Which is why I have not put much effort in to it, because it does not always make sense. Red and Blue would not necessarily be opposed to eachother-- fire and water to create steam for example. Clear and violet to give substance to mental constructs. White and black to sacrifice your own health to heal others.
And so on and so forth.
Thus, I think it works better simply as a representation on a larger sphere-- the ultimate goal of most wizards is to attempt to understand the mythical Eleventh Sphere, the sphere of Ascension where one literally becomes magic, or so the story goes (what actually happens I am not revealing here).
White <--Holy <--Relation to the Gods--> Unholy--> Black
Grey <-- Artificial <--Relation to Nature--> Natural--> Green
Brown <--Within <--Relation to the World--> Without--> Yellow
Blue <--Cold <--Relation to Heat Energy--> Hot--> Red
Clear <-- Physical <--Relation to the Mind--> Mental--> Violet
So, then, if the spheres do not represent mental dispositions, what do they represent? Abstract concepts-- each one represents a different section of the four dimensional hypersphere that is the Eleventh Sphere. Every element is, in some way, connected to every other element, but they are connected most strongly to their opposite. Thus why Brown has magnetic powers, and Yellow has meteoric summons-- they are intrinsically connected despite being, at the same time, opposites to each other. Brown is the power within the Earth, while Yellow is the power beyond the Earth, but both are inherently connected to the Earth.
An easier label of the pairings might go as such: Divine, Life, Source, Energy, and Substance, creating five colleges within the wizard's school of magic, a much easier number to organize than ten (which is why I did the pairing system to begin with).
The College of Divines: Studies holy and shadow magic. There are no demons or angelic hosts in this setting, but there are deities. The exact nature of these deities, however, is only known by the archmages. There are five archmages for this subdivision of the wizard's college-- two Holy archmages, two Shadow archmages, and one headmaster who oversees both. The headmaster of the College of Divines has been the grandmaster of the Wizard's College more often than any other headmaster.
The Campus of Twin Divinities looks much like a cathedral, with the white magic students cloistered in the upper levels while the black magic students study within the crypts below. Their attire tends towards traditional loose mage's robes, with either hoods or hats depending on the taste of the wearer.
The College of Life: Studies druidic and golemantic/technomantic magics. There is only one archmage, and only by necessity of needing a headmaster-- the most powerful golemancers prefer to call themselves Artificers, while the most powerful druids call themselves Grand Druids. The headmaster, however, still wields a lot of power within the college because of their bureaucratic authority, and thus they tend to be the one keeping both sides from fighting. Indeed, the head of the College of Life is often the one called upon to make decisions regarding diplomatic matters for the Wizard's College as a whole. The Campus of Creative Magicks Is made up of a widely disparate group of workshops, nature preserves, parks, zoos, arenas, and storage areas which are spread roughly in a circle around the central dormitory. Their attire tends to be more functional than other schools, even incorporating armor, although they all wear the clockwork tree symbol of the college somewhere on their robe.
The College of Sources: Studies geomantic and astrological magics. There are seven archmages of the order, three Astrological and four Geomantic, with the headmaster being the most powerful Astrologist. The reason for this splitting has been lost to annals of time, but the headmaster of the College of Source is still required to respect and run their opposing school, thus they tend to be long-lived masters of both, but rarely become the Grandmaster of the wizard's college. The Campus of Heaven and Earth is what is traditionally thought of when commoners think of Wizard colleges, striking as it does both deep in to the ground and in to the sky, with Geomancers studing in the rocky cellars while Astrologists peering at the stars from the tower above. Their attire tends to be more showy than the rest, including jewelry created by Geomancers, arranged in patterns that are said to be fortuitous by Astrologists, and because of the necessities of the campus, they tend to wear pants and tunics, believed to be more suitable for climbing stairs as the various initiates have to do until they learn their a travel magic.
The College of Energy: Studies pyromancy and aquamancy. There are three archmages, whom are rotated out amongst the top fifteen wizards of the college. The headmaster is more of a military general than anything, and indeed may not even be a wizard at all. This often results in scorn from the other schools, but the wizards of the College of Energy always give their headmaster respect, for they have almost always earned it. As a result of the separation of the Grandmaster and Master Militant, the headmaster of the College of Energy is almost never also a grandmaster, instead holding what is often the more powerful position of Master Militant. The Campus of the Elements is often described as a glorified barracks and military camp, and there is some justification for this despite it having a combination of pools, lakes, fire pits, and bonfires. The headmaster runs the college with the efficiency of a general, and they train their wizards in more physical combat as well as magical. As a result, their attire often includes chainmail and a sheathed (and heavily enchanted) weapon, or even a magical musket traded from the College of Life.
The College of Substance: Studies general arcana and illusions. There are ten archmages, evenly split between the college, with the two most powerful sharing the headmaster's spot unless one of them concedes it. In the case of concession, the headmaster of the College of Substance rivals the College of Divinity in number of grandmasters, although this is happening less and less as time goes on. The Campus of Mind and Matter is a wondrous sight to behold, a floating castle surrounded by wondrously resplendent energies and images, and it is known as a party school. Despite this, it provides the best general education to its students of any of the colleges, and is by far the most egalitarian and creative-- even if its students often take longer to complete its courses than in other colleges. Their attire tends to fit in with the fashion of the times, varying as much between decades as between individual students, but they always wear a circlet which glows when worn by an approved member of the college-- and unwanted guests may find themselves being pushed off to fall to their doom!