Congress made trouble for President Ronald Reagan in his last two years by investigating the Iran-contra scandal, overriding his veto of highway legislation and rejecting his Supreme Court nominee Robert H. Bork. In Mr. Bush’s winning campaign, however, he used “liberal Democrats” on Capitol Hill as a foil for his “read my lips, no new taxes” pledge at his nominating convention in New Orleans.

If she were the 2016 Democratic nominee, Mrs. Clinton would have a similar opening to deride Tea Party Republicans in the House even if Democrats keep the Senate this November. But all-Republican control of Congress would magnify it.

In part, of course, that depends on what Republicans do in the next two years. Mr. Obama has been frustrated by his inability, even after a decisive 2012 re-election victory, to “break the fever” of hard-core Republican opposition. With control of both chambers of Congress, Republicans might feel a greater responsibility to join Mr. Obama in governance by striking politically appealing compromises.

“If the direction Republicans define is where the country needs to go, then the initiative is with Republicans,” said David Winston, a Republican strategist for House leaders. “If Republicans just define themselves as being opposed to President Obama, then Republicans hand the initiative to Clinton.”

But evidence from recent years offers little reason to expect that the party will move in a more popular direction. The power of the Republican right has prevented the House, for example, from acting on an immigration overhaul or avoiding a government shutdown despite the express wishes of Speaker John A. Boehner.

Congress as an institution suffers from approval ratings about 30 points lower than Mr. Obama’s. The Republican Party fares worse nationally than Democrats — and has almost continuously since the latter years of President George W. Bush’s administration.

“A Republican Congress will present an inviting contrast and easily understood negative for whoever runs for president as a Democrat,” said Anita Dunn, a former White House communications director for Mr. Obama.