WASHINGTON — By the end of last week, it certainly looked as if Barack Obama had outsourced his presidency to Bill Clinton. First, he cut a Clintonian-style deal with Republicans on tax cuts and then he literally turned over the White House lectern to his predecessor.

Equally riveting and astonishing, Mr. Clinton’s blast-from-the-past performance in the White House briefing room on Friday afternoon reinforced the impression of political déjà vu, the sense that once again a Democratic president humbled by midterm elections was pivoting to the center at the expense of his own supporters.

But as no less an authority than Mr. Clinton reminded us, the comparison is incomplete and imperfect. “The story line is how well we worked with the Republicans and all that,” he said during his brief West Wing comeback. “But you know, we played political kabuki for a year.”...