THE ENDURING PUZZLE of modern politics involves a bit of magic - how do invisible people make themselves seen and heard?

The black minority figured it out. Now it appears that another vast constituency, long neglected, not listened to much, is beginning to learn the trick: the 11 million Hispanic-Americans.

"Hispanics are learning how the game is played," said Alfonso Ludi, an equal-opportunity officer at NASA by day and a community activist on his own time.

Willie Velasquez, of San Antonio, who heads the Southwest Voter Registration Education Project, explains: "It's similar to what happened in the South with blacks. The same thing is happening with Latinos except we're a couple years behind. But there's no question that political action is the priority now."

Rep. Edward R. Roybal, the Los Angeles Democrat who has started a fledgling five-member Latino Caucus on Capitol Hill, puts the transition in this perspective: "I still think we're in a gray area. I don't think you could say the Hispanic community is being recognized yet, but the potential is there. There's a different attitude."

While these things are impossible to measure, Washington does have some tangible evidence that the citizens of Spanish origin, from Mexican-Americans to Puerto Ricans, are influencing the political decision-making with more force and more sophistication. This does not mean the millennium is at hand - any more than blacks have arrived at their political objectives. Still, the change is evident from a few years ago when the wide array of Hispanic groups often squabbled among themselves and watched in frustration as other special interests moved in on the pie - especially federal jobs and fund.

Consider these scattered examples:

The heat is rising on President Carter to follow through on his generous campaign promises of jobs for Hispanic-Americans, whose disappointment ranges from mild to furious. While some groups work with a velvet glove of low-key persuasion, others are keeping their "frustration visible," as Ludy put it.

This month, the Hispanic-American archbiship of Sante Fe., N.M., the Most Rev. Robert Sanchez, is scheduled to celebrate a special mass at the Lincoln Memorial, followed by La Marcha de Reconocimiento - March of Recognition - around the White House. Ludi, an organizer, accused the Carter administration of lapsing into "the business-as-asual treatment of our needs as soon as the election ends."

Some Hispanic spokesmen are confronting, on a number of levels, the traditional priorities of filling minority-designated jobs with blacks.

"Generally, I think of our goals are similar," said Lupe Saldana, an equal-opportunity officer at the Environmental Protection Agency and Washington representative for the GI Forum. "But we do find that history has put the blacks in almost of our middleman, where we have to go to him for our money and our jobs, and we don't like that particularly."

The Mexican-American viewpoint, which has become much more unified in the last two years, is now an important factor in whatever legislation organized labor hopes to attain on the problem of illegal aliens from Mexico. The combined force of Hispanic groups might not be strong enough to sell all of their own proposals, but it is acknowledged that they can ally themselves with other interests to help kill the legislation, as they did last year.

The Hispanic groups went head-to-head with the NAACP last year on a particular issue - whether the Voting Rights Act should be broadened to cover the Spanish-speaking of the Southwest - and the Hispanic position prevailed in Congress.

The growth of Latino elected officials has been slow over the last decade, according to Velasquez, but he expects the pace to quicken, based on the voter drives underway. Mexican-Americans now claim two governors and 74 legislators in the five states of Texas, Colorado, Arizona, New Mexico and California where they have 17 per cent of the population. Only 6 per cent of municipal officials are Spanish-origin citizens.

"There is a new type of leadership that is sophiscated and moderate," said E.B. Duarte, a special assistant and Hispanic liaison for the U.S. immigration commissioner. "These aren't people who are off the wall. They're talking government jobs and doing a good job with them. It's not different really from the Irish or Italians or blacks. It's the American way." A Federal Beachhead

A DECADE AGO, when minorities were in ferment and struggling to develop stronger voices in politics, the only Hispanic figures who got much attention nationally were the ones with the most provocative rhetoric, staging street confrontations or worse in the interest of arousing their own people and building political movements. In a sense, their success is reflected in the greater sophistication and more substantial political muscle which has eclipsed them.

One place where Hispanics have made a small but important beachhead is in the federal government itself - the "ghetto in public service," as Ludi called the equal-opportunity offices in federal agencies. This at least gives them access to make the complaint, to gather the data and churn the political system.

Their favorite statistics are these: from 1968 to 1975, a period when minority employment was supposedly a priority, the federal government increased its Hispanic jobholders only from 2.8 to 3.3 per cent. They expect Carter to do better, faster.

On the other hand, Ludi said, access and prodding to change things, as he has learned at NASA. The agency was burned by controversy when its well-known civil rights officer was fired. In the year which followed, Ludi said, equal-opportunity hiring improved considerably - including 33 Hispanic engineers, slightly more than the number of black engineers hired.

The black-brown conflicts keeps popping up. Manuel Fierro, president of El Congreso, a lobbying coalition, wrote a very strong letter last month to HEW Secretary Joseph A. Califano Jr., opposing the appointment of William Robinson as director of HEW's Office of Civil Rights. Robinson is black and general counsel of the Equal Employment Opportunities Commission, where Fierro claims he neglected Hispanic employment discrimination.

"We've got to overcome this idea that civil rights positions are only for blacks," Fierro said. "They called us, trying to get us together, and I said, look, I'm not interested in trying to sensitize him any more. He ought to be sensitized already."

El Congreso has been shot down on some other initiatives. Fierro lobbied in vain for Hispanics to serve as EEOC chairman and as assistant secretary of state for Latin America.

On the other hand, the Carter administration has made at least three Hispanic appointments to jobs at the assistant secretary level - including the new commissioner of immigration and naturalization, Leonel Castillo of Houston. So Fierro and other leaders are generally optimistic about the future even while they continue to raise complaints. Playing "Catch-UP"

ON MOST ISSUES, the Hispanic organizations see a natural alliance with blacks, seeking greater funding for federal programs, protecting human rights in domestic settings. So the points of conflict are often minimized.

"Within the last several years, the Mexican-American community has been playing a kind of catch-up in many communities," said Manuel Lopez of Los Angeles, a leader of the Mexican-American Political Association. "At that time, they may have come in conflict with blacks who were in the managerial positions, but on a broader scale we seek coalition."

Rivalries between organizations and leaders are still a factor but apparently less dramatic than they once were. Congressman Roybal and othes are organizing a National Association of Latino Democratic Officials, hoping to pull together some 5,000 to 6,000 elected and appointed public officials as a nationwide assembly which would lobby government.

Legislation to curb the influx of illegal aliens by imposing federal penalties on employers who hire them has been bottled up in recent years by what Duarte calls "an unholy alliance" among the agribusiness lobby (which likes the influx of cheap labor), Hispanic-American groups, the Catholic Church (which supports the Hispanics) and civil libertarians who fear broad-brush police actions.

The concensus position of the Hispanics now is that they would go along with tougher contrls only with these conditions: The government grants as amnesty to the 8.5 million "undocumented workers" (5.5 million are Hispanic) who are already living and working in this country; the government creates some sort of reliable identity card for job-seekers so that Mexican-Americans will not be discriminated against; the government launches a long-range trades-and-aid program for Mexico to solve the problem at its source.

While the prospects for settlement are mixed at best, the Latino positin suggests what some see as another future issue - when Hispanic-Americans develop a heightened sense of foreign-policy interests just as other ethnic groups, Jews and Irish and blacks, have tried to help ancestral homelands through U.S. diplomacy.

That development is still a ways off. But these things do change. "It's not really very strong right now, but it's growing," said Saldana. "We will begin to take positions on things like the Panama Canal treaty. We're gathering momentum. We're beginning to see a broader picture."