Tuesday night’s dinner may have been the most consequential one yet in President Barack Obama’s quest to cultivate a more collaborative and collegial second-term relationship with Congress.

And the president pulled it off by, in essence, crashing one of the most quietly powerful, and rare, bipartisan social gatherings in the capital: the meal shared every month or so by the 20 women of the Senate.

Rather than traipse out to the suburban Virginia home of Republican Lisa Murkowski of Alaska, whose turn it was to host, the president invited the group over to the White House at the last minute — but asked if they could have the Alaskan halibut the senator had already arranged to ship in for the occasion. (The provenance of the peach pie was not disclosed.)

All 16 Democrats and four Republicans showed up, even though a couple had initially begged off because of scheduling problems. And, in keeping with the ground rules for their regular suppers, none spoke to reporters when the two-hour gathering broke up just before 9 p.m. The president’s press office said the group discussed the budget impasse, Obama’s job creation agenda, his proposal for federally funded universal preschool, the growing momentum for the bipartisan “gang of eight” immigration overhaul, last week’s defeat of his gun control agenda and the federal investigations and prosecutions in the Boston Marathon bombings.

For a couple of reasons, the meal held as much potential to benefit the president’s agenda as any of his earlier senatorial soirees.

Most tangibly, Obama’s guests control more legislative firepower than the clusters of senators at his three previous gatherings; eight Senate committees are currently led by females. Beyond that, there is a growing appearance the 20 are cultivating the sort of genuinely collegial, non-ideological, professional friendships that have become close to extinct in recent years — the sort of bonds that, in the eyes of so many veterans of the culture of Washington before the 1990s, were essential to making legislative compromise the norm rather than the exception back in the day. In addition, there is some research to support the “men are form Mars, women are from Venus” notion that female politicians are more regularly driven to achieve consensus than their male counterparts.

The guest lists for the other meals were assembled mainly in search of senators willing to compromise; five female Democrats were at the Jefferson Hotel supper a week ago, and three women attended Obama’s two meals with GOP senators.

Separate invitations to the president for one of the female senators’ dinners had been extended in recent months by Murkowski and Kirsten Gillibrand, D-N.Y. The ritual was the initial brainchild of the longest-serving woman in Congress, Democratic Sen. Barbara A. Mikulski of Maryland.

The seven chairwomen in the room were: Appropriations’ Mikulski, Budget’s Patty Murray of Washington, Agriculture’s Debbie Stabenow of Michigan, Indian Affairs’ Maria Cantwell of Washington, Intelligence’s Dianne Feinstein of California, Small Business and Entrepreneurship’s Mary L. Landrieu of Louisiana and Barbara Boxer of California, chairwoman of both the Ethics and Environment and Public Works committees.

The other nine Democratic senators are: Gillibrand, Tammy Baldwin of Wisconsin, Kay Hagan of North Carolina, Heidi Heitkamp of North Dakota, Mazie K. Hirono of Hawaii, Amy Klobuchar of Minnesota, Claire McCaskill of Missouri, Jeanne Shaheen of New Hampshire and Elizabeth Warren of Massachusetts.

The four Republicans are Murkowski, Kelly Ayotte of New Hampshire, Susan Collins of Maine and Deb Fischer of Nebraska.