[Read the Justice Department’s letter to Congress, along with the letter to Mr. Trump from Mr. Barr.]

There are few legal precedents about where to draw the line between Congress’s oversight power and the president’s authority to keep information secret. Past disputes have largely been resolved through negotiations and accommodations, so the matter never reached the Supreme Court.

But the fights across a range of fronts — including the president’s tax returns, how some of his associates obtained security clearances and underlying evidence from the inquiry by the special counsel, Robert S. Mueller III — have raised the prospect of litigation that could be appealed all the way to the highest court. Such rulings could provide a clearer understanding of the law, but the fight could go on so long that it would last beyond Mr. Trump’s term.

The House voted on Tuesday to authorize the Judiciary Committee to seek a court order forcing the executive branch to comply with two subpoenas related to the Mueller investigation, and it explicitly empowered committees to file such litigation over other subpoenas without votes of the full House. But it has not voted to hold any Trump official in contempt.

The fight centers on liberals’ suspicions that the census question could be a ploy to tilt the once-a-decade reapportionment of House seats. The Census Bureau has estimated that including the question would lead to a 5.8 percent decline in response rates from noncitizens, which Democrats fear would favor Republicans during the drawing of new House maps and deprive some states of federal resources. Apportionment of House districts has been based on raw population, not the number of eligible voters.

“I want to know why people like Kris Kobach, with a résumé of voter suppression techniques, have their fingerprints all over the most sensitive census operations that we have as a government,” Representative Alexandria Ocasio-Cortez, Democrat of New York, said as a bitter debate over the contempt citations unfolded between Republicans and Democrats on the panel. “This determines who is here. This determines who has power in the United States.”