Last August, a U.S. Air Force jet carrying a high-level congressional delegation landed here and unloaded its passengers. After a refueling stop, the aircraft took off empty for Hong Kong, where it remained for 48 hours before returning, still empty, to pick up the American lawmakers.

The round trip had to be made, despite a worldwide fuel shortage, to satisfy the peculiar requirements governing the "unofficial" U.S. relationship with Taiwan since Washington shifted its recognition from the island to the Chinese mainland in 1979. The State Department had ruled that it would be "inappropriate" for a U.S. government airplane to park on Taiwan.

Since its formal recognition of Peking as the rightful government of China, Washington has taken pains to avoid upsetting its new relationship with the mainland's rulers while trying to protect its substantial commercial, cultural and security interests in Taiwan.

Taiwan's Nationalist Chinese leaders had hoped the balance would swing back in their favor in the administration of Ronald Reagan, an old friend who had pledged during his campaign to upgrade contacts with Taipei, stop the niggling rules that have evolved to appease Peking and even reconsider Taiwan's request for advanced weapons, which was turned down by the Carter administration.

Thus far, however, Reagan's friendliness has done nothing visible to strengthen U.S. ties to Taipei, and Nationalist leaders who were exuberant after his election now speak more guardedly of the prospects for diplomatic gain.

"It takes two to formulate a friendship," said Vice Foreign Minister Frederick F. Chien in an interview. "From our side, we are always ready for anything. But it takes two."

The studied Chinese politics of patience, however, gives way to the slightest expression of pique when questions turn to Taiwan's longstanding request to buy a new fighter jet from the United States -- a plea that has set Peking-Washington relations on a collision course since Reagan's election.

"This is one question (on which) we had hoped to get a prompt answer," said Chien, while noting that "we know this is not the number one issue as far as the American government is concerned. We would not want to be regarded as betraying a traditional Chinese characteristic -- patience."

In the three years since the U.S. "derecognition" of Taipei, Nationalist Chinese officials have undergone frequent tests of that characteristic as Washington has turned from a ready arms supplier and trusted partner into a sometimes aloof and fickle capital.

Despite the downgrading of its diplomatic standing and the disappearance of the substantial American military presence here, Taiwan has demonstrated self-reliance. As a result, its relationship with Washington has changed more in form than in substance.

Although the ochre-colored American Embassy that stood here for six decades is closed, U.S. interests continue to be represented by the American Institute in Taiwan, an "unofficial" agency set up by Congress to maintain relations "between the people of the United States and the people of Taiwan."

The institute operates out of the former quarters of the U.S. Military Assistance Advisory Group and acts like an embassy in all but name. Its staff of "officers" are retired or "separated" from the foreign service, but they will return with seniority rights intact after their Taiwan tours end. They are paid from funds from the State Department budget.

The officers perform the usual diplomatic activities. They monitor political and economic trends. They process as many visas today as the embassy did, although technically the travel papers are issued by the U.S. consulate general in Hong Kong.

For appearances, the institute has invented a new vocabulary. The political section is called "general affairs" and consular activities are handled by "travel services."

The Americans dodge invitations to "official" Taiwan functions and stay away from government offices. Instead they arrange "social" get-togethers with Taiwan officials at restaurants, private homes and on the golf course.

Even though members of Congress have visited Taipei since derecognition, U.S. executive branch officials scrupulously stay away. Even a Department of Agriculture swine breeding expert was not allowed to come to assist Taiwan's pig farmers in 1980.

"We try to keep as low a profile as we can," explained an institute member. "The last thing we want to do is create publicity that will get the American government in hot water with the other side."

Perhaps the institute's most important role is nurturing the still burgeoning business ties between Taiwan and the United States. Two-way trade increased by 53 percent in two years, making Taiwan the eighth largest U.S. trading partner.

Trade officials estimate that commercial exchanges between the United States and this island of 18 million people totaled $14 billion last year. By comparison, American trade with the mainland was about $5.5 billion.

Since diplomatic relations were downgraded, 51 U.S. companies have sunk more than $280 million in new or expanded plants. Most ambitious is a $200 million truck and bus manufacturing facility General Motors is building in a joint venture with Taiwan.

American banks have loaned large sums of money that have increased their exposure here to several billion dollars. U.S. banks with branches in Taiwan have increased from eight to 11 since the shifting of recognition.

"There's no place like it in the developing world," said a U.S. banker based Taipei. "The workers like to work. The government knows how to govern. There's little red tape and we just don't see much of a risk."

For businessmen who have tried to navigate the bureaucracy of the mainland, spending months trying to find an official with authority to make a decision, doing business with Taipei is a dream.

General Electric Co. needed only 14 days in 1980 to get approval to supply $18 million in turbine generators for Taiwan's nuclear power industry.

Even Taiwan's insecure political status causes little worry among bullish investors who believe they would be well positioned if Taiwan rejoined the mainland.

"This is the place to get a foothold on that huge market," said the American banker. "It could be the financial center for the whole (reunited) country."

What engenders such confidence is Taiwan's remarkable economic performance. Although the economy slipped immediately after the shift in relations, it quickly regained its rhythm. Since 1979, the real gross national product has grown at a respectable 7.37 percent on the average per year.

What impresses economists even more is the island's 16.5 percent increase in labor productivity since 1979 while real wages have risen only 10 percent.

Taiwan has sought to reduce its reliance on the U.S. market that accounts for about a third of its exports. The spurt in U.S. business activity since 1979 was only part of the doubling of foreign investment in the first two years after derecognition.

Taiwan has increased its two-way trade with Western European countries by 72 percent since the American official pullout. There were no European banks with branches in Taipei before derecognition; now there are six.

The Nationalist government in 1980 lifted a ban on trade with five East European countries -- East Germany, Poland, Czechoslovakia, Hungary and Yugoslavia -- and exchanged about $20 million worth of goods.

Diplomatically, Taiwan has managed to retain full relations with all but one nation -- Colombia -- that recognized it at the time of the American withdrawal. Twenty- three nations -- almost half from Central and Latin America -- still recognize Taipei as the legitimate government of China.

Taiwan's economy has prompted many nations that jilted Taipei diplomatically to creep back commercially. Several nations with embassies in Peking have strengthened their ties with Taipei by setting up trade or cultural affairs offices here to augment unofficial links.

"The Republic of China's position in the international arena is not shrinking, is not fading away," said Vice Foreign Minister Chien. "We are very much a viable member of the international community."

While its the economy continues to bolster its international status, Taiwan is not resting its survival on commercial ties alone.

Nationalist Chinese leaders have seen in the Reagan election a chance to recoup some of their diplomatic losses and to ensure the American commitment to their defense -- at least to the extent that that was made possible by the congressional act permitting U.S. sales of defensive weapons to the island.

Although Washington continues to sell between $700 million and $800 million worth of military equipment to Taiwan each year, Taiwan authorities are pressing for a new FX bomber they say they need to offset sophisticated aircraft Peking is now designing.

Taiwan officials and political analysts say the aircraft is vital for the island to maintain its confidence against the mainland and continue to attract foreign investors who want some guarantee of the island's future.

But other political observers here believe the request is as much a test of the American commitment to Taiwan as it reflects the island's real need for new weaponry.

"It has come to symbolize the American interest in Taiwan's future," said an American businessman stationed in Taipei.