DAMASCUS, SYRIA, SEPT. 2 -- Eleven months after being recalled home, U.S. Ambassador William Eagleton returned quietly to the Syrian capital tonight in the first symbolic American reward to Syria for closing the Damascus offices of the Abu Nidal terrorist organization and for what the Reagan administration sees as improved Syrian behavior in the troubled region.

Eagleton, a career diplomat and Arab specialist, left Damascus last October amid a storm of indignation in Washington and Western Europe over Syrian involvement in terrorism.

In a display of solidarity with Britain after court testimony there linked Syrian intelligence officials to the attempted April 1986 bombing of an Israeli El Al airliner at London's Heathrow Airport, Washington also imposed several economic sanctions against Syria last fall. Those are still in effect but "under constant review," according to State Department spokeswoman Phyllis Oakley.

In June, Syrian authorities shut down the offices here of the Abu Nidal organization, named after its leader, who was banished from the Palestine Liberation Organization in the mid-1970s.

The organization, whose official name is the Fatah Revolutionary Council, is believed to be behind several major terrorist acts, including the bombing of airports in Rome and Vienna in 1985.

There have been widely conflicting reports in recent years about the organization's leader, whose real name is Sabri Banna. His whereabouts, if he is indeed alive, are a closely kept secret. He was reported previously to have lived in Libya.

The shutdown of the organization's offices in Damascus has been a welcome sign of normalization of Syria's deportment, according to senior western diplomats here.

Despite official silence over Eagleton's return, sources here said, Syria is relieved he is back.

A Syrian official, who declined to be identified, said: "Syria is watching with great satisfaction the return to normal of U.S.-Syrian relations because they were abnormal for us."

"Syrians view this step as a big jump," one diplomat said. "The way they see it is that after this everything will be easier. What they don't realize is that other sanctions will not be lifted automatically."

U.S. economic aid was halted by Congress in November 1983. Other American sanctions, imposed late last year and still in effect, include import and airline restrictions.

Although questions remain on whether Syria has responded to western concerns about terrorism, "as long as there are positive signs, we will see more normalization," one diplomat predicted.

When the State Department announced last month that Eagleton would return, it said this was "the result of our assessment of the broader Syrian attitude, policy and behavior with regard to terrorism," and not linked specifically to the escape or release on Aug. 18 of American journalist Charles Glass, kidnaped two months earlier by gunmen south of Beirut.

Oakley said that despite the U.S. expression of appreciation for Syrian efforts on behalf of Glass, no cause-and-effect relationship could be established on the circumstances that led to his freedom.

West Germany took the lead in rehabilitating Syria diplomatically, sending a new ambassador here in June and unfreezing two major credit lines in July.

In another move clearly aimed at warming relations, Bonn last month revoked an arrest warrant issued last November against an aide to Syria's Air Force intelligence chief who had been linked by testimony in Britain to the El Al attack.

Spokesmen in Bonn said the decision to revoke the warrant was "political and not judicial."

Britain, which severed diplomatic relations with Syria over the El Al attack, has said it is awaiting more evidence of Syrian good will before it resumes ties. Greece's foreign minister has just visited Damascus, and a European Community representative is expected this weekend.

Western analysts say there has been no evidence since April 1986 of any terrorist activity that can be traced to Syria. It still remains on the list of countries that the U.S. government considers to be supporting international terrorism, however.

Syria is seen as important in the region because of its key role in any Middle East peace conference, its claim to act as a stabilizing influence in Lebanon and its ties to both Iran and Arab countries -- a relationship seen in the West as making it a channel of communication in the Persian Gulf conflict.