In most places, neoconservatives and hawkish liberals fight McGovernites and old leftists in intellectual salons and leave it at that. In New York, they carry their fights to the streets and the ballot boxes. The classic battle in the genre was the 1976 Democratic Senate primary between Daniel Patrick Moynihan and Bella Abzug. Moynihan was loved by the neoconservatives and a variety of other ardent anti-communists. He was loathed by the old left and the McGovernites, who backed Bella. Pat beat Bella -- the race was that personal -- partly because Ramsey Clark was on the ballot to steal a share of the gauchiste vote from her.

In 1980, it is the same war by other means, only the soldiers are different. The Moynihan stand-in in Bess Myerson, consumer columnist, adviser to corporations and the former Miss American who wore mink coats on "Beat the Clock." For Abzugites, there is Rep. Elizabeth Holtzman, Watergate Commitee Nixon-hater, and scourge of the Defense Department and the Brooklyn Democratic machine. Her critics insist she only learned to smile when she announced her candidacy last January.

The voters of New York can be forgiven for not realizing that they are being asked to play their parts in an ideological holy war. For one thing, the rhetorical skirmishes have been quite mild by Bella-Pat standards. Myerson has occasionally criticized Holtzman's steady stream of votes against the military budget, but neither of them has considered ideological combat the most propitious form of attack. And there is that man standing between them who gave each her first major policical job. John Lindsay, fiscal crisis be damned, is trying a political revival, and he has carefully positioned himself in the ideological no-person's land between his rivals. There is also a fourth candidate, John Santucci, a conservative Queens district attorney. His main function in the race is to allow political statisticians to argue about which of the main contenders he is hurting most.

All this fighting over a rather dubious prize, the change to face New York's heretofore undefeated Republican senior senator, Jacob Javits.

The first candidate to announce this year was Liz Holtzman, and that is not surprising. There is a stolid seriousness about her that even allies find disconcerting -- though they usually will add that this is her greatest virtue. In Congress, Holtzman has built an utterly consistent liberal voting record, and has consistently voted this is an example of her courage -- the only candidate in the race to speak out against all but the most carefully constructed business tax cuts. And unlike lesser liberals, she has exposed scandal in social service programs, and gotten them shut down.

But friend and enemy alike usually return to her first-in-her high-school-class personality. Like Bella, she is tough on staff. But the best analogy is the eminently serious Javits. "If you had to invent a Jacob Javits who was younger and female," said one New York politics watcher, "you'd invent Liz Holtzman."

It's not as if Myerson would win the race's Ms. Congeniality Award. She, too, can be difficult. It took media adviser David Garth a long time to decide whether he'd take on the Myerson campaign. But he did, and that has made a lot of difference. Myerson was charged by her critics with being unfamiliar with the issues, so Garth has had her litter her speeches with authoritative statistics, usually strange and unassailable numbers like 413. She was criticized by Holtzman and in some leftish quarters like the Village Voice for her ties to big business -- Myerson has consulted on "consumer matters" for a variety of large corporations. Garth decided to turn this into an asset. The Myerson slogan: "She knows government. She knows business. She knows how to get things done . . . for New York."

And she also has the support of the ever-popular Ed Koch, for whom she campaigned hard in 1977, and Daniel Patrick Moynihan, who reportedly cringes at the prospect of serving with Holtzman. Myerson's central campaign theme is the same as Moynihan's 1976 line: that New York has been consistently shortchanged by the federal government. The Moynihan staff has been feeding her ammunition, and the whole act comes off well, especially for someone who supposedly doesn't know the issues. Finally, years of service at UJA dinners -- not to mention the fact that Bess was the first Jewish Miss America -- have won her many friends among New York's Jews, who cast about one-third of New York's Democratic primary vote.

But if you want charm, your candidate is Lindsay. The best line out of Lindsay's 1965 mayoral campaign was Murray Kempton's: "He's fresh and everyone else is tired." No one in this race is particularly tired, but Lindsay is still fresh, oozing partrician self-confidence and easy-going warmth. When the candidates get together to debate, Lindsay always has a kind word for one and all. He also has a sense of humor, the sort that once led him to urge that WASP's be called ASP's, since all Anglo Saxon Protestants are white, so why be redundant?

A popular joke of the election is that support for Lindsay is directly proportional to an area's distance from New York City. That's partly true -- Lindsay has strong organizational support upstate and the backing of about two dozen upstate county chairmen. He also has friends in New York's black and Spanish communities, which still have fond memories of the Lindsay regime.

But Lindsay, whose administration lived in a state of war with the white ethnics who inhabit New York's outer boroughs, clearly learned something during his tenure as mayor, and he is trying hard to win over his former foes. A key element in his campaign is support for tax credits for families with children in parochial schools. In New York City, with its large population of Jewish day school students, that idea appeals beyond the Catholic enclaves. In support of the idea, Lindsay also steals from Moynihan: tax credits for private schools are a Moynihan idea.

Which brings us back to ideology, something that underlies all of this. When the votes come in on Sept. 9, Holtzman is certain to do well on Manhattan's liberal West Side, Myerson better in places like Queens. Holtzman's defense record is a major factor behind this. Early in the campaign, Myerson attacked Holtzman for being "one of a small minority of legislators who has consistently opposed all United States defense expenditures." She added that "if your vote was the only vote passed in Congress in terms of our defense policy, we would have no defense." Holtzman counters that Myerson's attacks are a distortion, and that her defense votes were against "too much political pork barrel" in the military budget.

"If everyone voted as I did on defense," says Holtzman, "we wouldn't have a B1 bomber or a neutron bomb. Secondly, we would have a more muscular and effective national defense because my votes have been votes against waste." This, of course, is a classic defense argument, and the race will be seen as a referendum on that subject.

The outcome, though, could end up hanging on money. Here Myerson has the advantage, even though she actually has raised less money than Holtzman. Part of Myerson's advantage is that she's rich, so she can kick in as much money as she needs. And she has been able to raise substantial sums from the usual givers of big money in New York politics -- real estate types, people in the investment community, and the like. Holtzman accurately describes the Myerson givers as "the city's power elite." The point is that raising big money costs less than raising small money, so Myerson has been able to put seven dollars of every 10 raised directly into television. She seems likely to spend $1 million on television before the primary's over. So far, she's raised $500,000.

The good news for Holtzman is that she's raised $800,000. The bad news is that she's done it through direct mail, and so has had to recyle a quarter of her take back into the mails. Bob Squire, Holtzman's media adviser, says he hopes to be able to spend $400,000 on the primary.

All this could be academic, depending on what happens to 76-year-old Jacob Javits. Most people seem to think Javits will defeat his primary adversary, Al D'Amato, the presiding supervisor of Hempstead, Long Island. D'Amato is running a straight-out conservative campaign against Javits, but Javits seems to have helped himself inside the Republican Party by his early capitulation to Reagan's nomination. And D'Amato lacks the flair or obvious intelligence of Jim Buckley, who won election in 1970, only to lose to Moynihan's six years later. Still, don't rule out D'Amato. The New York Republican Party is small, and quite conservative.

The place Javits is really popular is in the Democratic Party. A New York Times/CBS Poll of Democratic primary voters in April showed Javits with a favorable to unfavorable ratio of better than 2 to 1. If Javits were running as a Democrat, odds are he'd beat them all.