Could the House seek a court order on its own?

Yes, and the House may soon ask Judge Howell to issue an order permitting its members access to grand-jury information from the special counsel investigation. It is not clear whether Mr. Barr would then direct the Justice Department to remain neutral, or instead to oppose the request — and, if Judge Howell agrees to issue such an order, file an appeal of that ruling.

Whether anyone is in a position to appeal such an order is important because the Court of Appeals for the District of Columbia recently issued a ruling that put forward a narrow view of when courts may let outsiders to see grand jury information. Judge Howell, a former Senate Judiciary Committee staff lawyer, may be more inclined to find that Congress has a right to access such material.

What about executive privilege?

A valid assertion of executive privilege by the president can also provide a basis to lawfully defy a subpoena, so Mr. Trump’s invocation of it gives Mr. Barr an argument and a shield as the matter moves toward a likely court fight.

The Supreme Court has ruled that presidents have the authority to keep secret certain internal communications, including discussions with aides about how to carry out their constitutional duties. The lines are murky between where Congress’s power stops and the president’s begins, in part because such disputes are usually resolved through compromises rather than definitive rulings.

Notably, judges have said they want to see the two branches engage in good-faith negotiations to find a compromise that accommodates the constitutional needs of each, so Mr. Trump’s overt stonewalling may undermine the executive branch’s position when some of these disputes get into court.

But in the exchange of letters between the Trump administration and the Judiciary Committee leading up to Wednesday’s vote and invocation, the executive branch argued that it was trying to make accommodations in good faith about the Mueller report, so it was Congress that was being unreasonable.

The Trump administration had offered to let a small number of congressional leaders read a less-redacted version of the report that would permit them to see information related to, for example, ongoing investigations and cases, although not the grand-jury material.