President Carter's Cabinet shakeup took a bitter and bizarre turn yesterday as the president put out the word that Joseph A. Califano Jr. was lying about the reasons given for his being fired as secretary of health, education and welfare.

Califano retorted: "I accurately reported what the president told me. . . ."

About the only charge unchallenged in the case of Califano vs. Carter et all is that, in the end, the HEW secretary was fired because he had made enemies within the White House and had left a thick residue of feeling that he often worked to undercut all the president's men. Only the names and the facts were in dispute.

White House officials disputed point by point Califano's public version of the private meeting Wednesday night when Carter told him he was to be replaced. Califano had given his version of the reasons in a press conference Thursday.

Yesterday, after maintaining one day of silence despite repeated press queries, White House press secretary Jody Powell and other presidential assistants said Carter's position is that virtually none of what Califano said is true.

The president said, through his spokeman, that:

He never told Califano he was being fired because he could not get along with senior presidential assistants Hamilton Jordan, Powell and Frank Moore, Califano said Carter specifically had named those three men.

He never mentioned the 1980 election campaign as a reason for the Cabinet changes. Califano had said Carter told him he needed to make the changes to get his Cabinet in line for the 1980 election effort.

He never told Califano that he was the "best" HEW secretary in history, only that he had done a "good" or "competent" job of managing HEW. Califano maintains Carter told him that he was the greatest.

But, the president's assistants say on orders of their boss, Carter did raise some reasons for the firing with Califano when the two men were alone in the President's office Wednesday in a session that began shortly after 6 p.m. and went on for 15 and 20 minutes.

The president says, through a spokesman, he told the HEW secretary that:

Califano had not been able to lend his support to Carter's bill to create a separate Education Department. (In fact, White House officials feel Califano had been working against this measure, once a Carter campaign promise.)

Califano had not moved expeditiously on developing a more modest version of a national health insurance plan, despite repeated requests to do so for White House domestic policy chief Stuart Elizenstat. (It was, the aide says, Carter's only reference to Califano's differences with a staff member). Califano learned toward a more ambitious plan than the one Carter and his top staff aides favored.

Califano was not effective in helping to achieve enactment of administration legislation on Capitol Hill.

Word of the president's version of the reasons given Califano for his firing were relayed to the outsted secretary.

Califano's response was to say that he stood by his original account.

Two Democratic representatives, Charles Rangel of New York and James Corman of California had called Carter Wednesday to urge him to retain Califano. Rangel said Carter called them back Thursday and told them Califano had done an "outstanding" job at HEW, and that there was "no question of Joe's effectiveness." Corman recalls Carter saying that Califano had been an "excellent" HEW secretary.

But after all versions have been heard, the basic fact of the firing of Califano is that the president decided to back his top assistants in what has been a long-running series of bickerings and battles between the White House staff and the HEW secretary.

This is also true, apparently, in the case of the firing of Treasury Secretary W. Michael Blumenthal. According to a source close to Blumenthal, Carter began his fateful meeting with Blumenthal by saying he had done a "splendid job." But Carter reportedly said there had been serious friction between the treasury secretary and White House political assistants, so he had no alternative but to let Blumenthal go-even through one assistant, Eizenstat, had urged that Blumenthal be retained.

Califano had antagonized Hamilton Jordon, Jody Powell, Frank Moore and other White House insiders from the outset of the Carter administration.

At first there were clashes over personnel appointments. Califano reportedly did not clear several appointments with Jordan. And in one celebrated case on Capitol Hill, Califano pointedly refused to permit the only plum that the old Chicago political mchine of the late Richard Daley had asked after helping Carter to victory in 1976. The Chicago machine wanted to pick the man for the regional HEW job. Califano vetoed their choice -- then picked a nominee tied to the reform wing of the party.

Later, when Califano tried to name his undersecretary. Hale Champion, to become the head of the Social Security Administration, and to place a friend, Stanford Ross, in the undersecretary's job, Jordan forced the secretary to swtich the two.

More recently, Carter had become furious at hearing from members of Congress that Califano was not enthusistically supporting his bill to create a separate Education Department. Carter ordered Vice President Mondale to rebuke Califano for this, and it was done.

Later, Carter was displeased upon hearing that Califano was spending five days in Hawaii at the end of his trip to China -- at a time when the crucial vote on the education bill was near. Again, he had Mondale call Califano.

White House officials had let Califano's original version of his firing go unchallenged Thursday, according to Powell, because new chief of staff Hamilton Jordon had told the senior White House aides that the president had said if there was any criticism of outgoing Cabinet officials, the staff person responsible for the criticism would be fired. Jordon also said at that meeting that presidential aides should not express publicly criticisms of any facet of the way in which the Cabinet, subcabinet and White House shakeup has been handled, sources said.