France was treated to a remarkable sight today. In the center of the Cabinet room sat President Francois Mitterrand, founder of the modern French Socialist Party. Seated opposite was conservative Prime Minister Jacques Chirac. Around the polished, oblong table sat 36 members of France's new right-wing government.

The atmosphere was stiff and tense as television cameras panned around the room to record the launching of "cohabitation," the word used here to describe the unprecedented sharing of power between longstanding political enemies. There were no smiles, no polite converation. And when the 25-minute meeting was over, there was no traditional "family snapshot" of the new government on the steps of the Elysee presidential palace.

A week after the narrowest of conservative election victories, French political analysts disagree on how -- or even whether -- cohabitation will work. Some insist that the whole idea of power-sharing between right and left is so bizarre that it is doomed from the start. Others believe that both Mitterrand and Chirac have a common interest in the success of the experiment, if only because the electorate is likely to punish any politician who appears to be responsible for its failure.

For Americans, it can be difficult to grasp what all the fuss is about. The principle of the division of power with constitutional checks and balances is well accepted in the United States. As Secretary of State George P. Shultz noted yesterday after meeeting with both Mitterrand and Chirac, it is normal for a U.S. president to have to deal with a politically hostile Congress.

The U.S. experience, however, is misleading. What France faces is more than an institutional conflict between the executive and the legislative branches of government; it is a division within the executive branch itself. A better analogy would be to imagine President Reagan suddenly finding himself obliged to appoint a Cabinet full of Democrats answerable not to him but to a Democratic-controlled House of Representatives.

There is also the matter of the monarchical traditions of France's Fifth Republic, founded by Charles de Gaulle in 1958. Last week, Mitterrand enjoyed more power, in relative terms, than a U.S. president. Once he made a decision, it was accepted automatically by the government and National Assembly. It would have been regarded as beneath him to lobby legislators for support.

"All French presidents since de Gaulle have behaved like monarchs," said Samy Cohen, the author of a book about French foreign policy called "The Nuclear Monarchy." "The question now is whether Mitterrand will be able to adapt to a situation in which he can no longer act like a monarch."

This sharp reduction in the powers of France's "elected monarch," whose seven-year term does not expire until 1988, has provoked a debate here about whether the country has witnessed a change of republics. The new system has been variously labeled a return to the postwar Fourth Republic, when governments were made and unmade by parliament every six months on the average, or the de facto introduction of a "Sixth Republic" with entirely different constitutional rules.

Philippe Moreau Defarges, a professor of political science, believes that the next six months will be crucial to the fate of cohabitation.

"Chirac has taken a big gamble; he has until the fall to make cohabitation work," Moreau Defarges said. "The people who voted for him expect quick results. By the fall, the honeymoon will be over and a whole series of economic and social problems will have reemerged."

Adding to Chirac's problems is the fact that he has a tiny majority in the 577-seat assembly: just one seat, according to the latest calculations. His right-wing coalition comprises supporters of former prime minister Raymond Barre, who is opposed to the whole idea of cooperating with a left-wing president, and assorted independent deputies who may rebel against party discipline.

At today's Cabinet meeting, Chirac said his government would seek a vote of confidence from the National Assembly when it convenes for the first time on April 2. He also has announced that he will adopt legislation by decree in order to rush key points of the right-wing electoral program through parliament.

Because government decrees require the president's signature before they can become law, this in turn has raised the question of whether Mitterrand will go along with what amounts to the dismantling of part of the Socialist program without even the formality of a parliamentary debate. The legislation under consideration includes returning to private control industries and banks taken over by the Socialists in 1981, and the repeal of a new electoral system of proportional representation.

Political analysts point out that, while the president's powers may have been reduced sharply since last week, they remain significant. Mitterrand is still head of the armed forces. He can dissolve the National Assembly and delay legislation. He can call a referendum. In the event of "a serious threat to the institutions of the republic," he can assume emergency powers to rule as a virtual dictator under Article 16 of the constitution.

In practice, the government will be obliged to take Mitterrand's views into account even if the constitution gives it the right to "direct and decide the policy of the nation." This is particularly true in the field of foreign and national security policy, which traditionally has been regarded exclusively as presidential turf. Before the elections, Mitterrand took the precaution of placing his own men in dozens of key administrative posts; they will be difficult to dislodge.

While there is considerable consensus on French foreign policy, Cohen believes that the president and the prime minister easily could find themselves disagreeing sharply on how to respond to sudden, unexpected crises. "The constitution is contradictory. It says the president is head of the armed forces, but it also says that the government 'disposes' of the armed forces. If both sides interpret the constitution to the letter, there could be violent conflict," he said.

It is widely assumed that Mitterrand is anxious to prove his good faith by sticking to the rules of the constitutional game. The Socialist strategy is to allow the conservatives to take responsibility for running the country in the hope that public opinion will turn against them by the time the next presidential elections come around. In the short term, Mitterrand and Chirac have a joint interest in defeating Barre, the most popular of the right-wing candidates for president.