The way the Republicans tell it, Jimmy Carter played into their hands today. He brought "an empty platter" to a city ravaged by an 18 percent unemployment rate. He delivered an Edsel to an auto industry needing a shiny new Ferrari.

"I'm surprised he even dared come," said GOP National Chairman Bill Brock. "His visit emphasized his greatest failure -- an inexcusably high unemployment rate."

"It was like Sherman visiting Atlanta," said Reagan press spokesman James Brady. "He was returning to visit the scene of his crime.

But there was a heavy ring of rhetorical overkill in the words of the Republicans. For until today, they have found themselves in the enviable -- and unchallenged -- position of having a traditional Democratic issue -- jobs -- all to themselves. They've played it to the hilt in this traditionally Democratic, big-labor city.

In political terms, Carter's 5-minute stop here on his way to Tokyo was an attempt to regain that turf, a move that sparked more than a little envy among GOP spokesman over the power of the incumbency.

"It was a pretty good political maneuver to come out her at this time," conceded Sen. William V. Roth Jr. (R-Del.), who tried to stage his own media event with a midday visit to the neighboring auto city of Hamtramck. "Republicans being here gave him a national platform to attract attention," Roth said.

Republican leaders here this week have been sounding just like Democrats used to sound. They've talked about little else than jobs and how the economic policies of the Democrats in Washington have cost people their jobs all over the place.

"Republicans have been running around on Democratic turf," complained United Auto Workers union President Douglas Fraser. "I don't blame them. We left a vacuum.

Carter's action, Fraser and leaders of the big three auto companies agreed, was a good starting point, a skillful blend of symbolism and substance, timed to place the president on all the morning television news shows during the period when Republicans had hoped to have the stage all to themselves.

It was a display of presidential interest toward an industry that has long complained that it was overregulated and underappreciated.

Republicans, not to be outdone, reacted quickly. Almost before Air Force One had left Detroit Metropolitan Airport, the GOP's mimeograph machines were rolling, accusing the president of "hoodwinking Detroit."

"Mr. Carter has come to Detroit today and has said nothing," a statement drafted for Brock declared. "The administration has finally discovered what the people of Detroit have known for three years -- that Mr. Carter's policies have been a disaster both in human and economic terms. Today's trip is another example of his attempt to govern by style and not substance, by symbolism and not sound economic policy."

These were tough words, good old-fashioned political rhetoric of the "are you still beating your wife?" genre. But behind them was an obvious concern that Carter was stealing the Republican thunder.

"Just how successfully his proposals would play in November was very much in doubt. "This time it's a matter of who loses the election," Rep. James J. Blanchard (D-Mich.) said. "If Reagan keeps his mouth shut, it's going to be much harder for Carter to win back the loyalty of a lot of workers."

Carter seemed to offer a little to almost every special interest group today -- except the consumer. For workers, he held out continued emergency unemployment aid. For auto dealers, a bare-bones loan program. For UAW leaders, an expedited hearing on their complaints that the auto industry is being hurt by the import of foreign cars. For industrial giants, the speedup of a Treasury Department proposal that would allow them faster depreciation writeoffs and hints of easing government regulations. And for the industry as a whole, a tripartite auto committee comprised of representative of labor, business and government.

It was, the White House insisted, about as far as the president could go without new legislation. It was, Republicans said, a warmed-over set of proposals that many of their number had been advocating for months.

The moves held additional political liabilities for the president. By urging a speedup of the hearings on foreign imports, Carter put pressure on himself. Should the International Trade Commission find the UAW's charges correct, he may have to come up with a solution for them in the weeks before the November election. His hints of easing government regulations may alienate supporters on the left.

The Republican position on this clear. Both Reagan, the expected party nominee, and GOP Chairman Brock have called for a freeze on all environmental and mileage-per-gallon regulations.

"The auto industry is virtually being regulated to death," Reagan said in Michigan during the May primary. "It simply needs the freedom to compete, unhindered by whimsical, bureaucratic changes in energy, environmental and safety regulations."

Reagan wants less, not more, governmental involvement in industry, and he has come out against the Chrysler bailout -- a political sacred cow here.

He has been adamantly against highway safety regulations. "When Ralph Nader charges Detroit ignores safety, who thought of such things as headlights, taillights, brake lights, windshield wipers, four-wheel drive and disc brakes?" he asked in one recent speech.

But mostly, the Republicans would rather talk about unemployment than any complicated proposal from Washington, presidential or any other. It is a simple easy-to-understand issue, the kind elections are won and lost over. "We're going to talk jobs, jobs, all fall," Brock said in an interview today.

That's why he said Carter's trip here was a plus for the party. "He dramatized the jobs issue more dramatically than anything we could do, simply by showing up," he said.