IRAQ, NOT ALONE, broke off diplomatic ties with the United States in 1967 on the basis of a fabricated Arab charge that American (and British) planes had joined Israeli forces in the war against Arabs. The other day those ties were formally resumed, with the impulse on Iraq's side coming chiefly from its requirements in its continuing four-year war with Iran. The political arc thus described, from Arab rejection of Washington to Arab cultivation of Washington, is a useful reproof to the careless talk one often hears to the effect that American policy in Israel, Lebanon and so forth is continually undercutting this country's interests in the Middle East.

The fact is that for all of its numerous missteps and frustrations, the United States remains a more or less welcome part of the political scene in the area. It is seen as a principal economic player, as a patron of Arab regimes under one or another form of radical or Soviet pressure and as the one country with enough access and interest on both sides of the Israeli-Palestinian divide to become somehow a mediator. American policy often disappoints the United States' Arab friends, but the American role remains key. Iraq, which long denied itself the full benefits of normal relations, has just underlined the point.

It's clear enough why Iraq is back. It professes to be holding its own against Iran, in a war that Iraq started at a moment of Iran's evident internal distraction. But a huge foreign army sits on its soil, its shipping is exposed, and Iran's strain of Islamic fundamentalism is still a live threat. The fall of the shah, the chosen American favorite in the Persian Gulf, removed what Iraq saw as the fundamental anti-Iraq tilt of American policy, but, as far as Baghdad is concerned, there is much more the United States could do. Right now, for instance, it wants to draw Americans into further measures to isolate Iran and reduce its access to foreign arms.

To win a welcome in Washington, the regime has been willing to tone down some of the cruder aspects of its policy -- its open support of international terrorism, its use of poison gas and the special passion of its opposition to the existence of Israel. Torture of political suspects, however, goes on, international organizations report.

And what is in it for the United States? American diplomacy in the Middle East is in a phase not so much of grand plans as of feeling out new currents. With the Baghdad connection, the United States gets to identify more closely with an Arab cause -- the war against Iran; with the Arab regimes of Egypt, Jordan and Saudi Arabia, which are friendly to Iraq and the most open to Western ways; and with one of the few countries (Iraq) able to balance off Syria's bid for dominance in the Arab world. If the United States as yet has no clear strategy, then at least it is coming into a better position to play a useful regional role.