History calls it a tomb. Yet no mummy was ever found in it.
It is attributed to the 4th Dynasty pharaoh Khufu, (Cheops in Greek) yet the only reference to his name is upside-down in red paint on some quarry blocks discovered by sheer accident in an almost totally inaccessible pressure-relieving vault high above the so-called King's Chamber.
In fact, no inscriptions of any kind appear anywhere in the Great Pyramid.
And absolute proof is still needed - after some 4 500 years - to attribute its building to Khufu. More, that it is his tomb.
The one inescapable fact - and just about the only one - is that the pyramid exists.
This month Egypt's antiquities chief Zahi Hawass will dispatch his men up the sides of the pyramid seeking clues which, he hopes, will prove whether the so-called "air shafts" from the "Queen's" chamber deep inside this architectural wonder actually reach the outside.
If they do not, says Hawass, it will indicate another mysterious and unknown chamber within the pyramid.
In September a tiny robot, equipped with a video camera, crawled up one of the shafts to a strange, small door, discovered some four years ago.
While the world waited - the adventure was filmed live by National Geographic - the robot broke through ... to find yet another mysterious door blocking its path some 65m up the shaft.
All work was halted - and Hawass said he would consider his next move.
The men will now look for the shaft's exit point on the outside - if it exists.
Hawass's choice of words - if there is no exit, it will indicate an unknown chamber - mark an extraordinary change in his attitude towards people he once branded "Pyra-mad-ologists".
He now takes tea with engineer Robert Bauval (who says Egyptian civilisation is more than 12 000 years old: it's in the stars) and Graham Hancock (the constellation of Orion mirrors the Giza Plateau) and John Anthony West (the Sphinx was damaged by constant rain and floods 15 000 years ago), and entertains the theories he once found laughable.
Why the change?
The small door in the pyramid shaft. It should never have been there.
Hawass and the supreme council of antiquities agreed some four years ago to allow the first experimental robot up the shaft.
They didn't expect to find anything, putting paid to further schemes by Pyra-mad-ologists to probe what everyone knew was a mere tomb. But the robot found the door. To Hawass, who lives and breathes the Giza complex, it was an astounding revelation and a reminder that the complex still held secrets.
Academics had also raised questions about mysterious unknown secret chambers hidden within the pyramid.
In 1984, French architects Jean Patric Goidin and Gilles Dormion, part of a French archaeological mission to do an architectural analysis of the horizontal passage leading to the Queen's Chambe", found themselves amazed and elated. What they had found was extraordinary.
The language of their official report is somewhat dry - but the facts are fascinating: "There can scarcely be any doubt at this stage that beyond the walls of the horizontal corridor, there exists, by design, one or more cavities."
What they had noticed was a simple matter of different types of stone. The corridor has a step, and the stones that comprise it and surround it were of a different cut and size to the rest.
They said: "In December 1985 we advised the French Ministry of Foreign Affairs of the various unusual features we had observed in the Great Pyramid ...
"Microgravity tests were carried out in May 1986. They revealed an undeniable and significant variation of density ...
"The amplitude of the phenomenon detected implies a large cavity... In August, 1986, drilling by representatives of the French electricity board revealed, to everyone's surprise, a layer of fine sand."
The architects' conclusions, reported a year later: "There can scarcely be any doubt ... that beyond the walls of the horizontal corridor there exists by design one or more cavities.
"A main cavity is situated to the south-west of the corridor, corresponding to the maximum measurement by microgravity instruments ... This main cavity could link with a gallery running along the west side of the horizontal corridor ... a symmetrical gallery may even exist on the east side.
"Whatever the nature of these cavities - they are surely untouched; they have probably an important purpose as they are carefully concealed, a complex and heavy system used to mask or close off the access."
Modestly, the French mention that the technique they had used was a "first" in the field of archaeology.
Did this, too, convince Hawass that the pyramid held more secrets?
The Greek historian Herodotus catalogued Egyptian history, which was as old to him when he arrived at the Giza plateau as he is to us today. He remarks that he had been told that
the body of Cheops still reposed on an island below the pyramid, surrounded by a lake.
But the pyramid is built on a granite outcrop, and is remote from the river Nile.
However, architect Hans Hansen - another member of the French team - wrote that perhaps another, secret, entrance exists on the western side of the pyramid along the 35th layer of stones. (The main entrance is on the northern side.)
Also, he says, at a point 57,59m along that western base line, the passage mentioned by Herodotus may yet actually be found, descending to the Nile's low-water mark at the time of Khufu.
No more work has been done on the French findings - yet. But it is significant that the camera in the recent robot probe up the Queen's Chamber shaft "saw" fine white sand - just like that discovered by the Frenchmen in the horizontal corridor.
I visited the pyramid late on a hot Friday afternoon, and watched happy Egyptian family groups sitting on the huge stones in front of the main entrance or picnicking on rugs.
But I was about to fulfil a dream, and climbed up the narrow "new" stairs to the main entrance, and down into a long, sloping passage, bending over to escape damaging my head.
And there, inside, was the magic I knew would exist - a tingling, almost weightless light-headed electric sensation. Levitation came to mind and it was all I could do not to throw myself into the air and float. At the bottom of the sloping passage was an iron grille - cutting off access to the chamber far below ground. No one is allowed there anymore.
Minutes later I was outside the portal of the Queen's Chamber - where the Frenchmen had found their "cavity", and from where the tiny robot had clawed its way up the "air" shaft to the mysterious doors blocking its journey.
I was, too, at the foot of the immense Grand Gallery, with its massive corbelled roof and wooden staircase, leading upwards to the King's Chamber. Another forced genuflection to get beneath a huge stone in the ante-room, and I was there, alone, with the "sarcophagus" for which there is no explanation. Time had no meaning. I was light-headed with the Power.
Napoleon spent a night here alone. He would never, ever, talk about it other than to say the visions nearly drove him mad. Author Paul Brunton also spent the night alone in this chamber. When he did not emerge in the morning, a team went in search of him, finding him comatose and unable to walk, speaking of "great visions".
My plan to lie on the floor in the swirling, roaring silence was thwarted by a loud American who stumbled into the chamber, shouting "Goddamn!" Had I the power to curse, he would instantly have become a horned toad.
Outside again, the world continued. Wet with perspiration, I sat shakily on a huge stone and fumbled for a Camel, forcing the ethereal, whispering, swirling voices to recede in the comfort of the everyday. And disappear into the dust of the Giza evening.
The dream fulfilled - but not yet satisfied....