A month ago, thousands of Turkish troops invaded northern Iraq in a mystery-shrouded operation that appears to have been directed at helping shore up the Baathist government in Baghdad, which has publicly welcomed the limited invasion of its own territory.

Today almost nothing is known with any certainty about the curious operation, which may still be in progress despite official denials by the Turkish government, which has blacked out just about all news of the operation. The government asserts that its troops were simply searching out "bandits" attacking Turkish territory.

Arab opponents of beleaguered Iraqi President Saddam Hussein assert that Turkey's intervention was designed as a warning to Iran and to Kurdish tribesmen who have renewed their war against the Baghdad government that Ankara would not tolerate chaos on its eastern border. Iraq and Turkey strongly deny such assertions.

According to the theory of those who oppose the Baathists in Iraq, the Sunni Moslems who rule Turkey, Saudi Arabia and other conservative Persian Gulf states would fear a victory by Shiite Iran in the nearly three-year-old war between Baghdad and Tehran. This would probably leave power in the hands of Iraq's Shiites.

In any event, Turkey's military strike into Iraq would appear to underline a growing instability in Iraq produced in part by the uprising of the Kurds, a people who have no homeland of their own and who have constantly agitated for at least autonomy in Iran, Iraq and Turkey.

Ayatollah Ruhollah Khomeini's Islamic revolution in 1979 in Iran touched off unrest throughout the region, and in much of Iranian Kurdistan the Kurds took up arms and successfully defied the central government.

Iraq's invasion of Iran in September 1980 drew the Iraqi Army away from northeastern Iraq, encouraging the Kurds there to take advantage of the situation to renew their battle against the Baghdad government. While fighting each other, the central governments in Baghdad and Tehran also had to cope with the separate Kurdish uprisings.

And at about the same time, the Turkish military strengthened its tight control over Kurds in eastern Turkey, who already had been subject to martial law for years.

The turbulence throughout Kurdistan triggered separate Iranian and Turkish military operations along their frontiers with Iraq last month. The attacks into Kurdish strongholds came amid reports that a "liberated" Kurdish autonomous region was about to be proclaimed in northern Iraq.

In what Iranian Kurds described as a "vast government offensive," Iranian troops for the first time wrested control of key roads linking Sardasht, Baneh and Piranshah from the rebels.

Iran for all intents and purposes succeeded in sealing off long stretches of its borders with Iraq and Turkey.

The Turks moved May 25 against targets in difficult mountain terrain crossed only by unpaved roads near the Iraqi city of Zakhu not far from the junction of the Turkish, Syrian and Iraqi borders.

The only official account of the operation some 400 miles southeast of here remains a barebones statement from the Turkish Foreign Ministry. According to the statement, the operation ended June 2 and the death toll was one Turk and one "bandit"--a contention met with widespread disbelief here.

The government undercut its casualty figures in the third week of June by allowing the censored Turkish press to reprint a report from a New York-based Armenian publication lamenting the deaths of 22 members of the Armenian Secret Army for the Liberation of Armenia in the operation.

In addition to uncovering larger casualty figures than the government claimed, the report added another aspect to the operation. Until then the presence of the Armenian secret army only had been rumored. It is a mysterious but effective terrorist organization that attacks Turkish diplomats and other officials to avenge what it claims was the slaughter during World War I of more than 1 million Armenians and to help reestablish the ancient independent Armenian state on Turkish soil. According to the publication, the Armenian secret army has formed a tactical alliance with the Kurds, apparently trading worldwide Armenian connections and safe houses for Kurdish ground organization against the Turkish government.

The number of Turkish troops involved in the operation is still a matter of conjecture, with some persons contending that as many as 14,000 were committed while others insist that in such tortuous terrain efficient command and control ruled out the use of many more than 4,000.

A State Department source said late last week that the Turkish operation had lasted only about five days, at the end of May, and was now over except for the possibility of some "residual units" in the area. He put the size of the Turkish force at two brigades--which would mean a few thousand troops--and said he had no precise casualty figures. He stressed the U.S. view that Turkey's chief motivation was to eliminate its Kurdish opposition.

Recalling a 1979 Iraqi-Turkish agreement establishing a no-man's-land on either side of the frontier, government sources in Ankara insisted that this time Turkish troops penetrated only two miles inside Iraq. But Kurdish rebel sources continued to claim the Turks are still in Iraq, and up to 25 miles at times.

Turkey justified its intervention by citing the recent deaths of three soldiers and the wounding of an officer just inside the Turkish border. In the previous year more than 250 trucks plying the lucrative route to Baghdad had been waylaid or burned in the border area traditionally considered outside Ankara's and Baghdad's control and a haven for smuggling of arms, drugs and carpets.

There are indications that the invasion took Iraq by surprise, although the Turkish communique said Baghdad had fully agreed to the operation, which it described as limited in time and scope.

The Iraqi military attache here, for example, appeared surprised at the first rumors about an operation requiring excellent intelligence and careful and time-consuming planning, Western embassy sources said.

Turkish Foreign Minister Ilter Turkmen flew to Baghdad several days after the operation began and later Mahmud Kaysi, the Iraqi ambassador, said it was being conducted "in cooperation" between the two governments.

The Soviet Union, which had criticized the NATO exercises, said nothing. Moscow is again backing Iraq following the deterioration of their relations with Iran.

Iran, apparently because of its own concern over Kurdish nationalism, made no criticism at the start of the operation. When it did speak out it was more to belittle Saddam Hussein and his inability to police his border than to attack Turkey.

Privately Turkish officials insist that "we did not intervene to relieve pressure on the Iraqi government."

But when asked about the possibility of large-scale disorders and anarchy following any overthrow of Saddam Hussein, the same officials do nothing to discourage suggestions that Turkey would be obliged to act.

The problem is just what Turkey could and should do without disastrous repercussions if it sought to act outside its borders.

Apart from the potentially dangerous precedent the present operation has created for getting drawn into Iraqi affairs, reflective Turks wonder whether anything more than establishing a buffer zone lies within Turkey's power.

"Like the poor, Kurdish nationalism will always be with us," one diplomat said, "and smart Turks know it."

For want of solving their own Kurdish problem--somewhere between a fifth and a quarter of the Turkish population is of Kurdish extraction although they are officially called "mountain Turks"--the military regime has saturated southeast Turkey with troops.