So where’s the stuffed goat? Of course that’s what I was wondering as I trotted along the corridors of Freemason’s Hall. I mean, I may not know much about this secretive fraternity, but I’ve read about their odd rituals and their initiation rites, about their rosetted aprons, rolled trousers and knobbly handshakes. And hasn’t everyone heard those rumours that involve ungulates? It’s natural to be nosey. No wonder that, as soon as I heard that the society was staging an exhibition exploring the history of its British headquarters and throwing in a rare behind-the-scenes tour to boot, I immediately signed up.

Freemason’s Hall is quite a landmark in Central London. And it certainly looks a pretty intriguing place. A cliff-face of stone soars upwards from the low clutter of Covent Garden like the intimidating façade of some massive fortress. The great bronze doors that guard it seem to be always locked. So it might surprise you to know that if you want to broach it, you have only to slip round to a side entrance and wander straight in. This building is the headquarters of the world’s oldest secular fraternal society, an organisation that has about half a million members in this country. They meet in “lodges” scattered all over Britain, and foster affiliations with some five million other members overseas.

Nobody actually knows the origins of Freemasonry. Its roots are embedded in legends that date back to the building of King Solomon’s Temple: a project — or so the story has it — so enormous that the stone-masons who worked on it started to organise themselves into groups.

More realistically, it may trace its beginnings to the medieval era. Travel for anyone outside the court or the Church was highly restricted in those times. But stonemasons were roving craftsmen, “free” to journey from site to site, as they brought their talents to the building of the great cathedrals of Europe. And the guilds that they belonged to were supposed to have developed their own secret codes of recognition: signs that became essential to distinguish a skilled master from an aspiring apprentice in the days when there were no credentials and only the top master masons could read. By the 17th century, however, membership of these guilds was beginning to decline and so lodges began to accept non-practising members chosen from among the more influential sectors of society. As these “accepted” members began to outnumber actual masons, discussions at meetings moved away from the skills of the trade towards broader consideration of such subjects as moral philosophy and charitable endeavour.

But the story of modern Masonry began in 1717 with the founding of the first Grand Lodge in England: a central regulatory body that held its meetings in assorted taverns and City Livery Halls. In 1776 it began to build premises of its own and it is the architectural history of these that a new show — The Hall in the Garden — now explores. It looks at the original Grand Temple, constructed by Thomas Sandby (and later destroyed by fire); Sir John Soane’s greatly expanded redevelopments, and the building in 1931 of today’s enormous Art Deco monument presiding over its oddly triangular two-acre site.

This is probably the most complete Art Deco building in Britain. To enter it is to enter a world of polished marble surfaces and shimmering mosaics, of stained glass and painted stucco, of heavy bronze and dark wood. You wander along its interminable passageways. You pass the pompous (but not particularly interesting) paintings and sculptural portraits of past members: grandees who range from monarchs and Prince Regents to major aristocrats — and mostly kitted out in their frankly rather ridiculous regalia.

You go into the robing room and down the processional corridor along which the Masons process with their banners through successions of vestibules to the climactic Grand Temple.

It’s a bit like being on the set of The Da Vinci Code — in which the Freemasons, of course (how could Dan Brown resist?) take a part. The farther you go the more deeply you are drawn into a dense web of symbolism. Everything seems to mean something in this weird runic realm — but try to work out quite what and you may find yourself probing anything from long-lost Enochian and Noachide customs to the principles of ancient Sumerian science or Euclidean geometries.

The single fundamental belief that Freemasonry requires is a faith in a supreme being, in some “divine architect”. And their so-called Volume of Sacred Law is the bible. It is one of three objects, “the three great lights” (the other two being a square and set of compasses — the symbols of the stonemason’s trade) that must be displayed when lodges meet.

But apart from these motifs the profuse symbology seems pretty much pick’n’mix. Here is a serpent of wisdom. There the theological ladder (which led Jacob to his vision). There the 12 signs of the zodiac. Here a tassel representing some mystic tie. There the sprig of acacia, mark of the immortal soul.

From the bronze door handles (swords with their tips embedded because this is a society dedicated to peace) to the blue lino in the corridor (a practical and rather cheaper modern day equivalent of the lapis lazuli in Solomon’s Temple) everything has meaning. And everything, it seems, can be Masonic, from a silver toast-rack (with prongs shaped from compasses and feet like little squares) to a jelly mould in the shape of a seven-pointed star (representing the seven liberal arts). Pretty soon your imagination starts to run amok.

A black-and-white chequered carpet is laid on the floor of the Grand Temple. Apparently one visitor was sure that he had an explanation — it was the hat band that runs around a policeman’s cap. But then, after a bit you will probably believe anything — even that Jack the Ripper was a Mason, as some like to speculate (regardless of the fact that no one has discovered who the Ripper actually was).

Disappointingly, I spotted neither hide nor hoof of a goat. Perhaps you will be content to make do with a pug dog instead. There are plenty of those because, apparently, when the Pope excommunicated the German Freemasons in 1736 they continued their after-hours activities under the name of the Mopsorden (The Order of the Pug) and made lots of little figures in Meissen porcelain.

But what in the end is revealed by this rare behind-the-scenes tour? Nothing, so far as I could find out, except that we all have a ridiculous fascination for mysteries and that men who wear pinnies probably prefer not to have to explain their proclivities. I can’t impart any code words or recount any rituals or tell you the right way to roll down your socks.

But I can assure any Art Deco enthusiasts that they will have a fascinating trip and all budding cryptologists will find plenty to cut their teeth on. Apart from that, probably best to bear in mind the English Masons’ own motto: Vide, Aude, Tace. See, dare, be silent — and most especially about that goat.

The Hall in the Garden is at Freemason’s Hall (020-7395 9257, or www.freemasonry.london.museum) from Thursday to October 20