Hardly anyone in Alexandria is sure why it was built, what it was meant to resemble and for what purpose out-of-town vistors arrive at its imposing brass doors by the busload and disappear inside.

The George Washington Masonic National Memorial, the 333-foot-high granite tower that the nation's largest fraternal order, the Masons, maintains on a grassy hilltop just inside the Beltway, is Alexandria's most visible landmark but without a doubt its least understood.

There are those who say that Masons, being Masons, prefer it that way.

The men who run the structure, dedicated 51 years ago, scoff at that. It exists not to foster Masonic mystery but to do proper honor to George Washington, "the most prominent, the most outstanding Mason this country has ever produced," says Marvin Fowler, the memorial's executive secretary-treasurer.

Despite its forbidding visage, the memorial is open to the public from 9 a.m. to 5 p.m. Inside is one of the country's most elaborate collections of George Washington memorabilia and, for anyone who wants it, a crash course on the centuries-old Masonic esoterica.

Not everyone knows Washington was a Masonic worshipful master, overseeing meetings of Alexandria Lodge 22 in an Old Town tavern, laying the cornerstone of the U.S. Capitol building with silver trowel and Masonic ritual, receiving a Masonic funeral on his death.

To assure his role in the order is remembered and to safeguard the "relics" of his lodge from fire and misappropriation, Masons around the country banded together early in this century to erect a memorial.

On Nov. 1, 1923, the cornerstone was put into place atop Shooters Hill, seen as an auspicious spot because it was once considered as the site for the Capitol Building itself. Calvin Coolidge, William Howard Taft and that same silver trowel were in attendance. Dedication came in 1931.

The memorial is, in fact, the only joint endeavor of the clannish American Masons, whose affairs are governed by 50 independent grand lodges answering to no supreme body.

The order claims 3.5 million members in this country today.

Members generally trace the order's origin to the stone masons' guilds of the Middle Ages or the ancient world. Initiates are versed in complex, and to a large degree still secret, litanies in which the plumb, the level and other tools of the mason's trade assume moral significance.

Washington--one of 14 U.S. presidents that Masons claim as members--once said the order's principles promoted "private virtue and public prosperity" and it is in abstract terms like these that modern-day members continue to explain its reason for existence.

Masonry has been criticized as segregationist (blacks and whites generally belong to separate lodges), anti-Catholic (numerous popes have forbidden Catholics to join) and a waste of good money and praised as a molder of men and endower of children's hospitals. One reason it endures may be because it satisfies a fundamental human need to belong, to organize for organizing's sake.

Each time an American in 33 states becomes a Mason, he is required to send $5 to the memorial, where a staff of 35 maintains the expansive, grassy hills around it, tends a large collection of Masonic literature and shepherds through echoing halls any visitors who show up.

The memorial also draws on donations and income from an endowment to fund its $500,000-a-year budget. Money is adequate but never abundant, says Fowler. Workmen are slowly completing a $75,000 job of replacing broken storm windows that mar the structure's appearance from the outside.

The building was modeled after the fabled lighthouse of Alexandria, Egypt, one of the seven wonders of the ancient world. It is built of granite blocks--some measure 20 feet long--and reinforced concrete and capped by a double keystone, one of the countless objects in Masonic iconography.

Its proprietors also point out, with no small pride, that it has two of the world's few slanting elevators, running 7 degrees off the vertical.

Visitors step into the richly marbled Memorial Hall to find a 17-foot bronze statue of a standing Washington. He is arrayed in a Mason's ceremonial apron, and around his neck wears his "master's jewel," the symbol of authority as head of a lodge.

Nearby is a replica of Washington's lodge room, with his "master's chair" from Mount Vernon and his actual Masonic apron. Also on display in the memorial are the Washington family Bible, a pocketknife he is said to have carried for 56 years, the instruments with which doctors bled him on his deathbed and a clock that was stopped at 10:30 p.m. by an attending doctor to mark his death on Dec. 14, 1799.

On the floors above are elaborately decorated rooms dedicated to the major Masonic subgroups. The Knights Templar have a Gothic chapel with stained-glass windows. The Royal Arch Room is done up in Hebraic style and contains a depiction of the Ark of the Covenant, the vessel in which the Ten Commandments were kept. Visitors get a brief glimpse of it as ceremonial curtains open and close automatically.

On the lower levels, the Shriners honor their order's luminaries. There are portraits of smiling leaders in red fezes and a motor-driven phalanx of miniature horsemen, soldiers and marching musicians who move to recorded music in front of a model of the Taj Mahal.

All of this draws thousands of Masons each year from around the country on "pilgrimages." For some it is a truly spiritual experience. Others are simply curious, wondering what their money has gone toward all these years.

Recently, 72 members of the Masonic lodge in State College, Pa., and wives rolled up to the doors in two buses, making the lodge's first pilgrimage in 27 years.

Lodge Master Donald Boller recalls stepping into Memorial Hall and being stunned by its marble splendor and the presence of Washington. "The guy who was with me said, 'what's the matter, Mr. Boller?' because I stood with my mouth open and just looked."

Following tradition, Boller's lodge convened a meeting and awarded Masonic degrees in one of the memorial's two stately furnished meeting rooms. Using them are a treat for those Masons accustomed to the lodges in the American hinterland, which are often over drug stores or next to gas stations.

Though the memorial's prime function is to honor Washington, it is also the meeting place of his old lodge, which was renamed Alexandria-Washington after its first worshipful master. It now has about 1,000 members, many of them drawn, according to Master Lawrence Callahan, by its historic credentials.

Masonry was once a potent political force in Alexandria, with Masons holding seats on the City Council. Today, as the city has expanded beyond Old Town and younger people moved in, the memorial and the citizens have drifted apart.

"They see it from the outside and that's it," comments Anne Morris, vice president of the Cameron Valley Civic Association. "I don't think they even know what it is." Mayor Charles E. Beatley estimates that only one in 10 of the city's people has ever entered the memorial.

It is a thing that is simply there and, to many, is a towering eyesore. "It serves as a landmark," says Vice Mayor Jim Moran, "albeit an unattractive one. Ugliest thing."

It stares King Street drivers hard in the face, punctuates the views from countless Alexandria windows and gives pilots heading for National Airport a visual reference point on the ground.

"You say, 'abeam of the temple, tower,' and he clears you to land," says Mayor Beatley, a retired airline pilot who recalls sighting the memorial (it is topped by red beacon lights) on his first flights into National in 1943.

Its 36-acre grounds are one of the city's largest open spaces and draw guests the Masons do not always welcome. In winter, children flock in to sled on its grassy slopes and tales of fierce groundskeepers have created a perception among many that Masons are not the best of neighbors.

Fowler says sledding is not forbidden. It is just not specifically allowed, due to the liability implications if anyone should get hurt.

The Masons are also down on dog walking and have chased off unknown numbers of pet-owners, including Del. Marian Van Landingham. "My men that have to mow the grass, they don't like it at all," says Fowler.

Neighbors are more sympathetic with the Masons' attempts to rid themselves of carousing teen-agers, who periodically use the memorial's steps and winding approach road for late-night rendezvous, leaving behind beer cans, wrappers and other jetsam of adolescent good times. Last year, someone mutilated three crape myrtle shrubs and started a bonfire with the cuttings.

The Masons' property is tax-free, a fact not lost on city budget planners as they try to squeeze more money from the tax base. Officials at times gaze longingly at the huge site, situated next door to the soon-to-open King Street Metro station, and wish it were open to taxable development.

But the city must content itself with dollars spent by those the memorial draws into town (a shopping trip in Old Town for the wives is a standard part of the Masonic pilgrimage.)