A day after a bitter standoff over a judicial appointment between Gov. Terry McAuliffe (D) and the Republicans who control the Virginia General Assembly, both sides continued to disagree about the outcome.

The House of Delegates stood by its claim that Monday’s special legislative session remained underway Tuesday — even as McAuliffe and Senate Democrats dug in on their belief that it was all over.

The Senate’s abrupt vote to adjourn Monday — a legal maneuver intended to keep McAuliffe’s Supreme Court pick on the bench — caught Republicans off guard and sent them scrambling for legal research to challenge the move.

By Tuesday, House Republicans were roaring back with state constitution citations that they said backed their position. They also kept the flag atop the House chamber, where it flies only when the legislature is in session. The central flag on the Virginia Capitol also waved, as always. On the west wing, above the Senate chamber, the flagpole was bare.

That the House and Senate cannot even agree if they have adjourned shows — as plainly as the lopsided flag display — how off-kilter political discourse in Virginia’s capital city is right now.

The governor called a special session of the legislature to draw new congressional districts, as ordered by a federal court. No mapmaking took place. Senators and delegates instead spent the day fighting over who should fill a seat on the Supreme Court of Virginia. They chose no one. And the biggest coup of the session? Abruptly calling it to an end.

Led by Senate Democrats and made possible by one Republican defector, the surprise move to adjourn has left the fate of a sitting Supreme Court justice in question and has likely thrown the task of redrawing the commonwealth’s congressional map to a panel of federal judges. The only certainty is that the most baldly partisan brawl of McAuliffe’s governorship remains in full swing.

“The language of Article IV, Section 6 of the Constitution of Virginia is clear, unambiguous and emphatic: ‘Neither house shall, without the consent of the other, adjourn to another place, nor for more than three days,’ ” Speaker William J. Howell (R-Stafford) and Senate Majority Leader Thomas K. Norment Jr. (R-James City ) said in a joint statement. “By adjourning . . . without seeking the consent of the Virginia House of Delegates, the Lieutenant Governor and the Democratic leadership in the Senate of Virginia violated the Constitution.”

A spokeswoman for McAuliffe (D) did not answer directly the question of whether the legislature remains in session, instead criticizing Republican legislators for not taking the redistricting process more seriously.

“It is embarrassing that Republicans, who shrugged off the opportunity to move this process forward, are now engaged in an existential argument about whether or not they are in session,” spokeswoman Christina Nuckols said.

Nonetheless, the adjournment dispute could have real consequences for Justice Jane Marum Roush.

Republicans saw the session as an opportunity to try to unseat Roush, a highly regarded former Fairfax County Circuit Court judge. Republicans have largely praised Roush’s qualifications but objected to the process: McAuliffe did not consult the Republican leaders of the House or Senate before making the appointment.

The legislature normally makes judicial appointments in Virginia, but governors have the power to fill vacancies when the General Assembly is not in session. Those recess appointments expire 30 days after lawmakers convene unless they elect the judges to full terms.

After legislators refused to consider Roush for a full, 12-year term Monday, McAuliffe said he would give her a second interim appointment when the first one runs out in September. But he does not have the power to make that recess appointment if the legislature is in session, as the House insists is the case.

As the House and Senate dug in Tuesday, new details emerged about the administration’s last-minute discussions about delaying the session that jeopardized Roush. A rank-and-file member of the House said a senior administration official approached him about serving as a go-between for the governor’s office and Howell on the subject of delaying the session.

Del. J. Randall Minchew (R-Loudoun) said in an interview with The Post that the official called him Saturday afternoon, and they discussed whether the special session could be called off — at least until partisan tempers had cooled.

Minchew declined to identify the official, saying he did not want to break a confidence, but Nuckols later identified her as Suzette Denslow, McAuliffe’s deputy chief of staff.

“The basic conservation was, what could be done to relieve some of the tension and to allow for the governor and the legislature to work together in good faith toward electing a justice and dealing with other items of legislation on the docket for review,” Minchew said.

During the discussion, Minchew said the two concluded that the best way to do that might be to put off the session. He said he agreed to contact Howell to ask about the possibility that the House, Senate and governor would jointly call for a delay.

Nuckols had said in an interview Sunday that the administration never considered calling off the session but said some people had made that suggestion to the governor. On Tuesday, she said Minchew proposed the idea to Denslow.

She said Minchew asked, “ ‘Would you consider this?’ and Ms. Denslow said the only way the governor would consider that is if the Republican leaders made that request.”

Minchew disputed the notion that he proposed the idea, though once it came up, he said he volunteered to call Howell with it.

“The senior administration person called me,” he said. “I did not suggest that idea. We were talking conceptually that’s what would need to happen.”

If Howell had been agreeable, Minchew said, the administration official wanted the speaker to call the governor about it. The request is notable given that the fight over the judge turns almost entirely on what Republicans consider McAuliffe’s breach of protocol by failing to consult them.

Minchew said he called Howell to alert him about the outreach from the governor’s office. After consulting with other Republican leaders, the speaker decided it was too late. Delegates had canceled family vacations and made reservations here for the session, then just 36 hours away.

“He was appreciative that someone had reached out, albeit through a back-channel communication,” Minchew said.