Can a lifelong Socialist with four card- carrying Communists in his Cabinet and a compulsion for government ownership of banks and big business find lasting happiness in close alliance with a fiercely anti-communist champion of private enterprise?

On the answer hangs not just the future state of Franco-American relations but a large part of the Reagan administration's grand strategy for meeting the Soviet threat to Western Europe. So it matters whether the things that unite Ronald Reagan and France's President Francois Mitterrand outweigh the things that divide the two men almost diametrically in their social and economic ideologies.

And, to an astonishing degree, they do. Or so it would appear from just about every appraisal here, American as well as French, of the vibrations from the first meeting between the two men, in Yorktown last month.

There is almost a mirror-image in some of their similarities. As one French observer of the Yorktown scene puts it: "Both are mavericks, in a sense; both took a long time getting where they are; both have extreme solutions to comparable problems of inflation and unemployment; both are roughly of the same generation."

And both share a common perception of the Soviet threat, an aversion to communism, and a conviction that the Western alliance needs to rebuild and modernize its defenses to match growing Soviet strength.

True, the new administrations here and in Washington have almost as large a difference in their approach to the so-called Third World as they do in their domestic political philosophies. But even here, French diplomats take comfort in the ability of the two leaders to "agree to disagree."

There is, in short, almost an April-in-Paris quality to the current state of the relationship between the United States and France, far exceeding the conventional estimate of how things would go when Mitterrand was swept into power last May.

So much so that officials on both sides question seriously whether it may not be too good to last-- whether the same quality will still be there when April actually rolls around. The answer rests in large measure on yet another striking similarity between Reagan and Mitterrand: both have made their foreign policies hostage, to a perilous degree, to the success of radical socioeconomic revolutions whose prospects, at the moment, would appear to be problematical.

The Mitterrand government, for example, is predicting economic growth next year at more than double the present rate--even while its policies are visibly drying up private investment. An American expert finds that estimate "wildly optimistic" and a flimsy foundation for any economic policy. Even the predicted rate would not be enough to absorb a new surge of young entrants into the labor force.

Already, unemployment in France is at 1.8 million. It is expected to reach 2 million by the end of the year. That particular number is widely held to be a "psychological watermark." Reaching it would touch off strong, perhaps even violent, repercussions--and heavy pressure from the Socialist Party's left for even deeper government intervention, greater protectionism and an easing up of the hard anti-Soviet line in order to promote more job- creating trade with the communist bloc.

In short, a worsening of the troubled French economy could force a reappraisal of those elements of Mitterrand's policy most dear to the Reagan administration: the tough line against the Soviets, including French support for the deployment in Europe of American theater nuclear forces; and the steady buildup of French military strength.

Any significant shift in East-West policy would come hard for Mitterrand. His policy has broad popular support. The Socialist antipathy to communism runs deep; so does Mitterrand's. "He was a junior minister at the time of the Berlin Blockade," says one French official, "and has not forgiven the French Communists for thwarting his earlier efforts to win the presidency."

French pride in its independent military power and technology further reinforces the Mitterrand line. But economic imperatives will almost certainly take precedence if the Socialist economic revolution turns sour. The question, then, is not whether Mitterrand can find happiness with a practitioner of Reaganomics. The question on which the Franco-American relationship will almost certainly turn is whether he can find happiness with his own brand of economics. To find out, tune in six months or so from now.