But a close-in examination suggests that her reversal was an act more of self-preservation than of favor-trading, driven by intersecting motivations: avoiding the wrath of term-limited Council members who could undermine her speakership, distancing herself from an embarrassing scandal over City Council budgeting that had damaged her own mayoral prospects and, above all, protecting a political identity that hinged on a working partnership with the popular Mr. Bloomberg.

The year it unfolded, 2008, brought two political shocks that altered the course of both of their careers. Mr. Bloomberg, after flirting with a presidential candidacy as an independent, realized that the White House would never be his and turned his gaze back to New York City. Ms. Quinn, a Democrat who was the first woman to be Council speaker and who was just building a name beyond her West Village district, faced a federal inquiry into her office’s oversight of millions of dollars in taxpayers’ money.

Mr. Bloomberg’s needs drove what happened next. After the mayor decided he wanted another four years in City Hall, his aides expected Ms. Quinn to get it done, treating her almost as an afterthought. They assured Mr. Bloomberg, aides said in interviews, that she would abandon her own plan to run for mayor in 2009 and push through legislation allowing elected officials to serve 12 consecutive years, instead of 8.

When the mayor first spoke with Ms. Quinn about his designs on a third term, during a phone call in the fall of 2008, it was to tell her of his decision to rewrite the law, not to seek her approval beforehand, she said in an interview last week.

Ms. Quinn did not object. Asked about her reply to the mayor, she said, “I didn’t have much of one.”