Above a cloistered courtyard, hidden behind office blocks in the City of London, two men in suits stand on a Persian carpet and gaze out of the window.

In the centre of the courtyard, metal steps lead up to a large bowl. From below, three separate streams of water shoot up, a three-part liquid arc, combining and pooling in the bowl.

“Look at the fountain,” says Michael Jeans, settling into his winged armchair. “There are three spouts. And there are three ways to join the Haberdashers’ Company.”

Mr Jeans is talking specifically about the Haberdashers, the livery company of which he is past-master and now chair of the education committee. But much of what he says could apply equally to most of the London livery companies that have links with more than 160 schools around the country.

These organisations have a growing influence in English education as they have firmly involved themselves with the Government’s academy programme. This 160 figure is only going to rise, and as such it is worth delving into the mysterious world they inhabit.

The first of route to membership is the most straightforward: patrimony. The child of a freeman of the company has the automatic right to join. “That’s a right: it cannot be denied,” says Mr Jeans.

Next to him, Rear Admiral Richard Phillips, the company clerk (a sort of chief operating officer), interrupts. “Actually, you have a right to apply,” he says. Mr Jeans nods: “So it can be denied. But it ensures continuity.”

Then there is apprenticeship. On the basis that most company members’ knowledge of haberdashery does not stretch beyond the third floor of Liberty’s, this is not about learning a trade. Instead, outstanding head boys or head girls from Haberdashers’ schools are selected to join.

The final route is redemption, a term Mr Jeans acknowledges is redolent of born-again prayer meetings. “It means you don’t have to do a term of apprenticeship,” he says.

“Those are redeemed.” Thus redemption is used as a reward for Haberdasher headteachers, or as a way to bring in someone whose talents have been deemed desirable by the company.

Off-hand, Mr Jeans cannot recall the proportion of liverymen coming through each route. “It’s like I can’t tell you how many males and females we’ve got,” he says. “It’s not an issue. It’s about whether you’re a Haberdasher.

“We’re not in competition. Someone once said, ‘Do you have more schools than the Mercers?’ We don’t know and we don’t care. We’re not interested in being better - probably because we know we are. We’re just interested in good governance.”

Originally, livery companies were essentially medieval trades unions: merchants of a particular trade formed guilds to protect their own interests. Today, there are 108 livery companies, ranked in a strict order of precedence. Mercers are at the top, followed by Grocers and Drapers. Haberdashers are at number eight. When the order was settled, in 1515, the Merchant Taylors and the Skinners disagreed over which should be number six and which number seven. The solution was to alternate on an annual basis; this is believed to be the origin of the phrase “at sixes and sevens”.

The newer companies - the Worshipful Company of Information Technologists or the Worshipful Company of Hackney Carriage Drivers - still retain a professional allegiance. Most of the medieval trades, meanwhile - mercery, bowyery, brodery - have long ceased to exist. This makes little difference to the day-to-day running of the livery companies, however. When Elizabeth I wrote to the Mercers to ask why silk was so expensive, the company replied that it had no idea: by the 16th century, none of its members was actually a mercer.

What they do, instead, along with other charitable ventures, is run schools. There are 164 schools nationally with links to livery companies, new or old. “The four pillars of livery companies are: trade, education, charity and fellowship,” says Mike Jenkins, clerk to the Information Technologists’ Company (ITC). Granted livery status in 1992, the ITC now supports a south London school specialising in technology and sponsors a west London academy jointly with the Mercers.

“In the middle ages, the reason to get involved in education was to train people for their trade: to take young people and train them to be better goldsmiths, merchants, farriers. We’re still an organisation that believes that education of young people is protecting the future of our trade. The more qualified young people are, the more the British IT industry will benefit,” says Mr Jenkins.

Concealed in a side-alley in the City of London, Mercers’ Hall opens into a high-ceilinged ambulatory (“I call it an ambu-whatsit,” says the receptionist), complete with double staircase and stained-glass windows. In wood-panelled hallways, bearded and be-ruffed mercers gaze down from framed oil paintings. (Walking past similar paintings in his own hall, Rear Admiral Phillips comments: “Look at the cuffs and the ruffs! You can see why haberdashers made so much money.”)

Here in Mercers’ Hall, Michael Marchant, the company’s head of education, oversees involvement in 17 state and independent schools, including well-known institutions - St Paul’s in London, for example, and Thomas Telford in Shropshire - that dominate their respective league tables.

Both the Mercers and the Haberdashers have established a reputation for running successful academies; they take on new schools because local authorities approach them and ask them to make over one of their schools in their own educational image. “They come to us and say, ‘We’d like a Thomas Telford in our borough’,” says Mr Marchant. “We don’t go where we’re not wanted. You don’t want to force yourself on someone.”

For the Haberdashers, this involves fostering a traditional independent-school environment in each of their 13 schools, regardless of actual independence: there are houses, prize-givings, competitive sports and prefects in gowns. And each school will have company governors. “We’re totally convinced that good governors make a school,” says Rear Admiral Phillips. “It’s the board of governors that provides a sustaining environment. And, of course, the governors appoint the head.”

Haberdasher governors are given a passport-sized blue book, outlining their roles and responsibilities. They are also offered a training day and invited to a governors’ dinner. “What other rewards?” says Mr Jeans. “Well, I got a tankard when I finished as governor.

“It’s a bit like being a teacher: the reward will come through the overall esteem in which the Haberdashers’ Company is held, and the esteem in which the school is held. And the performance of the students.”

Mercers, too, work as governors for their company’s schools. “The company has the ability to direct resources,” says Mr Marchant. “If you live in leafy Gloucestershire and are committed to helping poor children, the Mercers’ work in and around London is a very good way to support that cause. We try to match talent up to where the work is needed.”

This is echoed by Mr Jenkins of the ITC. “If you’re an individual who wants to do good work, as an individual you can make a difference. But if you’re 10 people together, all doing good works, you can do so much more. By ganging together in the livery company, it’s an opportunity to make a real difference.”

The phrase “good works” is unavoidably redolent of Victorian do-gooders, benevolently toiling among deserving poor. Indeed, it is difficult to imagine inheritance of livery company membership being particularly common among pupils at any of the Haberdashers’ or Mercers’ own academies, many of which serve deprived catchment areas.

“It’s important to have an understanding of the state sector if you’re going to be a governor in a state school,” says Mr Marchant. “But that can be learnt. If people are motivated to help, then they’re motivated to go and find out. If people know what their role is, and they’re in it for the right reasons, I think a blend of internal and external understanding is quite healthy.”

“The head boys and head girls we’re bringing in do help to redress the balance,” says Mr Jeans. “Otherwise, yes, the majority would come from having an independent education. You’ll also find there’s some racial diversity among young liverymen, and that will eventually work its way through. But we’re not obsessed with being politically correct. I think we’re naturally politically correct.”

“That depends on the politics,” Rear Admiral Phillips interjects.

In Haberdashers’ Hall, he and Mr Jeans point out the traditional features - the Elizabethan charters, the stone coats of arms, the framed portraits - that line the corridors of the 2002 building. One of the rooms, complete with wood panelling and silver candleholders, was picked up in its entirety from the old hall, and transplanted into the new.

If there is a metaphor here, it is one that Mr Jenkins believes applies equally to his own company, the ITC. “Livery companies do have a thousand-year tradition, and they have a structure and a format that’s successful,” he says. “It’s an opportunity to get involved in these good works.

“In the entrance interviews for our company, the most-used line is, ‘I’ve worked really hard; I’ve done really well to get where I am now. Now I think it’s time to give something back.’ We provide a mechanism for doing that.”