TO SOME WASHINGTONIANS, the most intriguing tennis match played this spring was not at the French or at the Italian championships. It was a doubles match played by two senior White House aides, a senator and Secretary of State Alexander Haig.

Haig had been invited to play on the White House tennis court by Ronald Reagan's chief of staff, James Baker, who, along with Deputy Chief of Staff Michael Deaver, another player in the game, are two-thirds of the triumvirate that rules the White House staff. the fourth player was Sen. Paul Laxalt (R-Nev.), Reagan's closest friend on Capitol Hill.

Earlier in the year that foursome would have been an impossible combination. Haig was locked in a political wrangle with the White House over who would have primacy during foreign policy crises. Haig eventually lost his battle to Vice President George Bush, and then antagonized the White House staff with his public statements on the day Reagan was shot.

When the time came to make political amends, Baker, who managed Bush's presidential campaign, and Haig, the erstwhile antagonist, teamed up to beat Deaver and Laxalt 6-0, 6-2. It was a friendly game although there were some jokes that Laxalt, a good player, was miffed at having to play with Deaver, who is not up to the senator's level.

Thus was reinforced the political importance of the tennis court located on the most secluded spot of the grounds of the country's most closely watched building. Set on a low patch of land on the south lawn, the White House court is surrounded by shrubbery and fencing.

Tourists walking around the White House can't see the court from the sidewalk and would never know it is there. That is, after all, the point: to give a president, his staff and their guests some privacy in which to develop both lob shots and policy.

For the first time since 1974, the country is without a tennis-playing president. Gerald Ford, who took office in August that year, was an active and enthusiastic doubles player. The 1976 election brought in what may have been the highest-ranking tennis administration the country had seen. Both Jimmy Carter and Walter Mondale were tennis players who used the White House court. A hint of a tennis rivalry developed between the two, though they did not play often.