WE WANT OUR corporate identity to be as well-known as our subsidiaries," says Gordon Bowman, an officer of United Technologies Corporation. "We want to make ourselves visible."

As a result, UTC will make visible some of the finest paintings by old masters still in private hands: 57 items from the collection of Baron Hans Heinrich Thyssen-Bornemissa of Lugano, Switzerland will appear at the National Gallery from Nov. 17 through mid-February.

A modest line at the bottom of the traveling exhibit's lavish poster will say: "Made Possible by United Technologies."

UTC is the latest corporate entry in the international art show sweepstakes, whose participants include IBM (the Dresden and China shows); Exxon (King Tut and Thomas Jefferson); Control Data Corporation (the Hermitage show coming next year); Time, Inc. (Alexander the Great, later this year); Mobil Oil (Edvard Munch); Occidental Petroleum (three Russian shows); and Coca-cola (the Tokugawa collection of Noh Robes and Masks); among others.

The traveling exhibit game has three sides,and when it is played properly, theoretically everybody wins. On one side stand the governments with their national treasures. On the other, corporations with their bankrolls. In the middle are the museums, smiling enigmatically like the Mona Lisa.

And, of course, the cast of thousands - the people who waited in line to see the treasures of King Tut while others were waiting across town to see "Star Wars."

The purposes of all three participants are different, but by the time the visitors start strolling through the shows, the goals have usually been harmonized. For the museums and their visitors, the payoff is in spectacles unimaginable with limited resources - and, of course, the joys of contemplation. For the governments and the corporations, it is less tangible but no less precious: image, prestige, the momentary good will of a public that does not always think kindly of the Soviet Union, of Egypt, of Exxon or IBM or even Coca-Cola.

J. Carter Brown, director of the National Gallery, puts it gently: "the corporations are identifying themselves with excellence." Bowman is slightly more detailed: "We want to associate United Technologies with two things that are basic to our business: quality and fine craftsmanship." A curator from a midwestern museum not usually on the international exhibit itinerary calls it "vicarious class."

Whatever it is, it's expensive and sometimes risky. But in the past few years, sponsors have been lining up almost as vividly as the fans at the Tut trek (Mobil moved in quickly to subsidize the Munch show when another oil company lost interest), and their motives are generally clear.

Control Data, which will underwrite a lavish show from The Hermitage in Leningrad next year, hopes to generate enough dollars for The Hermitage to buy a Control Data computer system. Time, Inc. - whose exhibit "The Search for Alexander" is scheduled to open Nov. 15 at the National Gallery - is arranging a promotion modeled on a pop phenomenon of a few years ago, "The Ascent of Man." There will be a book (published by Time subsidiary Little, Brown) adn a six-part television series to be produced by Time-Life Films.

But aside from subsidiary rights, public relations and tax writeoffs, there are other possible motivations. Some museum staff members believe that IBM, for example, may see vast, untapped markets for business machinery in China and East Germany, though company spokesmen piously deny that they had any such thing in mind.

Moreover, the timing of international exhibitions frequently correlates with developments in international relations. The initial arrangements for King Tut and for the China exhibit ("Archaeological Finds in the People's Republic of China") both occurred during the shuttle diplomacy of Henry Kissinger, and both involved governments that were then moving toward normal diplomatic relations with the United States. The contract for Tut was signed three months after U.S.-Egyptian relations were resumed in 1974. The China show arrived a bit more than two years after Richard Nixon's visit to Peking. Three Russian shows have been brought here by Armand Hammer (alias Occidental Petroleum), who makes no secret of his interest in promoting detente. The coming Hermitage show was negotiated in the wake of the SALT II talks.

Greece, which long avoided such activity and even had a law to prevent the taking of ancient works of art out of the country, is suddenly breaking out with a rash of international shows: Cycladic Art, now at the National Gallery, Alexander, which will arrive in November, and "Greek Art from the Aegean Islands," which will open at the Metropolitan Museum in New York Nov. 1. And this flurry of activity happens just as Greece is entering the Common Market. (It may be relevant, too, that Greek Prime Minister Constantine Caramanlis, who made the Alexander show possible for years of dissappointment and delay, is a Macedonain like Alexander).

Brown points out that the time lag for the preparation of a major show (usually two or three years) would make fine-tuning difficult in a fast-changing diplomatic scene, but most international art shows seem to come from nations which are making a special diplomatic effort of one kind or another.

Whatever the reasons, Brown says that the National Gallery receives "a lot of pressure for various shows - from ambassadors, from scholars and from our own staff - many more shows than we can accomodate. With the East Building, we are now, like the Kennedy Center, able to run more than one production at a time, but we are still very new at the business of being a multi-ring circus."

The big shows draw phenomenal crowds, and a mixed response from curators. At the National Gallery, a close check is kept on attendance, and the blockbusters are fondly remembered: King Tut (7,025 per day); China (6,496 per day); Dresden (6,459 per day). "The Art of the Pacific Islands" is now attracting around 4,000 visitors per day - an excellent figure for a show which lacks corporate sponsorship and has had, therefore, relatively little promotion.

One museum director who scorns such figures is Lorenz Eitner, director of the Stanford Art Museum. Shows like Tut and Dresden, he said recently, "blot out everything else. They disorganize the museum totally. San Francisco shut down for four months in preparation for King Tut. Museums should be used thoughtfully by people who can profit from them intellectually and imaginatively - school children, the elderly, interested laymen - but not just crowds of people. The mass use of museums subjects them to physical strain and makes security a problem. Blockbusters pervert the nature of a museum, and turn it into something else."

If attendance figures are controversial, so are the ethics of handling corporate exhibits.Those battles are generally fought in private, and what debates do exist are smoothed over when the Gallery turns its face to the public.

The latest such controversy concerns "The Search for Alexander," which will be financed jointly by Time Inc. and the National Bank of Greece at a cost reported as high as $800,00 a piece. According to earlier accounts, the production of replicas from the show was supposed to be put in the hands of Alva Museum Replicas, Inc., a Time subsidiary. Several museums questioned this cozy deal, as well as the substantial marketing plan under which Time, Inc. subsidiaries would produce a book, a six-part TV series, catalogs and posters. Whatever may have been planned in the past, the arrangement now is one that the museum people (in many ways the "conscience" of traveling shows) can accept.

"I would like to see Alva get it," said Zachary Morfogen, who is handling Time Inc.'s interests in the show, "but it will be a competitive bidding situation. We want to be sure that whoever does it will meet our standards of quality, and Alva is used by many great museums. Control Data has picked Alva for the replicas of its Hermitage show." Morfogen is the managing director of Time, Inc.'s arts and books division, which includes Alva Replicas.

Whoever makes the replicas, the arrangement now is that any proceeds from their sales will go to the Greek Ministry of Culture - as the proceeds from the sale of Hermitages replicas will go to the Soviet Ministry of Culture. "We certainly do not expect to make a profit on the exhibit," says Morfogen, "although we do hope to have a return on the book."

Robert Amory, general counsel for the National Gallery, staunchly enies reports of severe in-fighting over what might be called the subsidiary rights to the show before Time agreed to the present arrangement. In negotiating the contracts for the show, he says, he was simply "protecting our interests as one museum, and the interests of all the museums that will exhibit the show."

The National Gallery's Brown will not deny that there are show-biz aspects to the phenomenon of traveling exhibits, but he insists that "everything we do is educational."

"Our permanent collection is focused on European and American paintings, with a few sculptures and graphics, from the late Middle Ages to the present - a very small proportion of the world's noteworthy art. We are primarily a picture gallery, like the National Gallery in London, the Prado or the Uffizi; it's a very different role from that of a local museum that takes a crack at everything, and we have a responsibility to offer the public things that supplement our permanent collection."

That supplement is perennially most successful at the National Gallery when it is offering what museum people call an "object show" (to distinguish it from a "painting show"). Painting shows have done very well at the National Gallery, and they get the lion's share of critical attention, but the blockbusters - Tut, Dresden and China - were all focused on objects.Next year's Hermitage show, sponsored by Control Data, will concentrate on objects but will have a distinguished set of paintings as a bonus.

And in the end, the giant shows clearly benefit the public. Many people are first attracted to a museum by an exhibit like Tut and than become regular visitors, a fact reflected in general attendance figures. Since 1968, when the National Gallery began to escalate its involvement in traveling shows, the annual attendance there has risen from 1.2 million to 5.5 million. And if the corporations bring in material so powerful that, in Eisner's phrase, it "blots out everything else," it is also true that people remember the Vermeer painting, the Chinese horse, the glittering accoutrements of Tut long after IBM and Exxon have faded from any association with these experiences.

This means two things: First, that the image-enhancing value of sponsorship is probably a short-range benefit and corporations will have to keep on repeating the process for any permanent return on their investment.

But it also means that the final winner in the international exhibition game, if there is to be only one winner, is the art. CAPTION: Picture, no caption, By Glenn Mosser for The Washington Post