President Reagan and his Democratic opposition came together in the White House Rose Garden yesterday in an unusual public show of bipartisan support for the administration's embattled $98.3 billion tax bill on the eve of a scheduled showdown vote on it in the House.

"All of us here are united by something much bigger than political labels," Reagan said, as Speaker Thomas P. O'Neill Jr. (D-Mass.) and other House leaders looked on soberly. "We are all Americans. We all want to get the American economy moving, and we all realize to achieve this we must cut federal deficits to bring down interest rates and create jobs."

Despite the coalition across party and ideological lines that the administration has assembled behind the tax bill, which former treasury secretary William Simon supported yesterday as "a necessary evil," a stubborn and equally strange coalition continued to oppose the measure.

Its opponents in the House include a combination of liberal Democrats and staunchly conservative Republicans. Many of the Republicans were wooed personally by the president yesterday in a series of White House meetings.

After freshman Rep. John P. Hiler (R-Ind.) said at the conclusion of one White House meeting of 30 conservatives that he didn't think Republicans opposed to the tax bill should necessarily vote for it out of loyalty to their president, Reagan made what one of the congressmen described as an "impassioned" speech for the measure, saying it was a necessary and worthwhile compromise.

The 71-year-old Reagan pointedly noted that he had been fighting for conservative principles when some of those attending the meeting, including the 29-year-old Hiler, were in grade school. The president concluded his appeal for support by quoting Barry Goldwater at the 1960 Republican national convention: "Let's grow up, conservatives."

Goldwater made the statement while urging that his name be withdrawn from consideration as a presidential nominee so conservatives could unite behind the candidacy of Richard M. Nixon. But yesterday, Goldwater made a speech on the Senate floor opposing the Reagan-supported tax bill.

As pressure mounted on opponents of the bill who continued to resist the leadership of both parties, confidence grew within the White House that it would be passed today. Officially, the view expressed by deputy press secretary Larry Speakes was that the administration was "still behind and making progress." But one administration official said he expected Democrats to provide the margin of victory even though a majority of Republicans still oppose the tax bill.

"The vote here remains up in the air," said Rep. Dick Cheney (R-Wyo.), chairman of the Republican Policy Committee. He added that anyone who said he had a hard count "has been smoking something."

But the leader of Cheney's party in the House predicted victory today.

"We're going to win," said House Minority Leader Robert H. Michel (R-Ill.), one of the congressional leaders who later joined Reagan in the Rose Garden. "I know we're going to win. I've got that visceral feeling."

Democratic leadership aides said their vote counts showed 75 to 100 Democratic House members on each side of the measure, with support very soft. But there was some feeling that the Democrats may have been downplaying their own support for the bill in an effort to keep the pressure on Republicans.

One Democratic aide said many liberals who would normally support a measure that combines tax reform with an extension of unemployment benefits were threatening to vote against the bill because they were angry at the president. "They think they've been getting beaten up for 18 months by Reagan and they'll be damned if they're going to save his ass on this one," the aide said.

Though conservative opponents of the measure claimed to have a vote count showing 206 members against, 126 in favor and the rest undecided, one opponent, Rep. Ed Bethune (R-Ark.), said, "That count is at least two days old." It was clear that divisions over the tax bill breached both party and ideological lines. In the California delegation, where Republicans have been especially virulent in their criticism of the bill, the count among Democrats as of yesterday afternoon was 10 to 9 against, with the state's four black congressmen opposed.

Rep. Henry A. Waxman (D-Calif.), a supporter, said he sensed erosion on the bill, and Rep. Norman Y. Mineta (D-Calif.), another supporter, said many Democrats would refrain from voting until they saw at least 100 Republican votes on the House tally board.

That would exceed earlier White House estimates of likely Republican backing. One official said yesterday the administration would be pleased with 80 GOP votes; another said 90 Republican votes were likely.

Beneath the facade of bipartisan cooperation, there also was the more customary political partisanship of a year in which the midterm congressional elections are 2 1/2 months away. One administration official, after observing that the White House was depending on Democratic votes "which we don't know how to count," added: "But I'll say this: If the Democratic votes aren't there, we'll have one helluva campaign issue."

At the White House yesterday, the president and his top aides worked hard, right through a Reagan dinner with 28 reluctant Republican congressmen, to convince doubters. Speakes released a letter from Teamsters president Roy Williams and a telegram from 20 other labor leaders supporting the bill.

A group of 35 business leaders who called themselves the "deficit reduction action group" met with Reagan in the Cabinet Room and announced their backing for the legislation. And John A. Albertine, who heads a group of fast-growing companies called the American Business Conference, said that while a tax increase was difficult to support five weeks ago, "it's now a joy."

Reagan's speech in the Rose Garden late yesterday afternoon was followed by another appeal from Simon, a secretary of treasury in the Nixon and Ford administrations, who presented a letter signed by him and 10 other former treasury secretaries supporting the tax bill. "The budget is not only out of control, it is hemorrhaging," Simon said. "This is necessary, a necessary evil if you will. I wouldn't support all parts of this package necessarily if I were here in Washington but I recognize that tradeoffs have to be made."

No one in either party professed to know how many votes the president actually changed, although Rep. Mickey Edwards (R-Okla.), head of the American Conservative Union, called Reagan the "most persuasive president in our history." But Edwards himself remained opposed to the tax bill.

Conservative southern "Boll Weevil" Democrats -- a key element in Reagan's earlier legislative triumphs -- also remained divided, despite a meeting of their own yesterday with Reagan.

The House vote is expected to be followed by prompt action in the Senate, where three additional Democratic senators, including Alan Cranston of California, yesterday joined Sen. Edward M. Kennedy (D-Mass.) in backing the measure. Without saying so publicly, the White House now regards Senate approval as a foregone conclusion.

One unmentioned factor that may be working in favor of the administration is the desire both of Congress and the President to begin their vacations. Congress is scheduled to start its recess Friday and the president, once the vote is taken, may leave town ahead of the congressmen en route for a two-week stay in California.