It isn’t often that the swearing-in of a new mayor of New York draws national television attention, but then, it isn’t every day that you see a mayor sworn in by a former president of the United States with a prospective presidential candidate also on the stage.

So there was plenty of symbolism and more than the usual amount of politics attached to the formal inauguration of Mayor Bill de Blasio on Wednesday. Issues such as the prospects of liberalism in an ideologically divided country, the future shape of the Democratic Party and the political ambitions of Hillary (and Bill) Clinton all played out in front of New York’s City Hall.

De Blasio, now one of the nation’s most liberal elected officials, delivered an unabashedly progressive inaugural speech that closely tracked the themes of his “tale of two cities” campaign. It was the kind of speech not often heard in national politics since Bill Clinton redefined the Democratic Party as New Democrats.

The new mayor, who was the unexpected winner of his party’s primary and then won a landslide victory in November, sought to disabuse those who thought he would scale back his liberal ambitions once he faced the challenges of governing. To the crowd that sat huddled against the cold and to those watching on cable TV, he said: “Let me be clear: When I said I would take dead aim at the tale of two cities, I meant it. And we will do it.”

Outgoing mayor Michael Bloomberg squeezed into a front row that included both Clintons and New York Gov. Andrew Cuomo (D). De Blasio, who had run explicitly on a platform of changing course from the Bloomberg years, briefly thanked the man who has run the city for a dozen years, first as a Republican and later as an independent.

But that was mostly perfunctory. For the rest of his address, he promised to push New York to the left, as quickly and aggressively as his political skills will allow him. “We are called to put an end to economic and social inequalities that threaten to unravel the city we love,” he said. “And so today, we commit to a new, progressive direction in New York.”

It is that impulse that will make de Blasio, the first Democratic mayor of the city in two decades, perhaps the nation’s most closely watched mayor in the coming months.

As he well knows, he has only limited power to undo the inequalities in a city where nearly half the population exists at 150 percent of the poverty level or less. His proposal to tax the rich to pay for early-childhood and after-school programs faces resistance. His day-to-day decisions may reflect pragmatism as much as progressive ambition in the face of political obstacles and management challenges that will test him constantly.

No matter. As the New York Times wrote in its lead story in Wednesday’s paper, de Blasio’s election will turn New York into a “closely watched laboratory for populist theories of government that have never before been enacted on such a large scale.”

De Blasio has become a beacon to those in the Democratic Party’s progressive wing, who have often been disappointed by or disillusioned with President Obama and what he has done and not done in office. The progressives see few political leaders on the left — Sen. Elizabeth Warren (D-Mass.) is one exception — willing to give voice to their agenda. All that would be enough to lend significance to Wednesday’s swearing-in. But it was the added presence of the former president and former secretary of state that gave it significance beyond the city’s boundaries.

The tableau on the stage spoke to the interrelationships among the New York political elite as well as to the evolution of the Democratic Party from Bill Clinton’s first presidential campaign to the present.

Cuomo is the son of former New York governor Mario Cuomo, who was a hero to the party’s liberal wing and whose decision not to seek the presidency in 1992 opened the door to Clinton’s victory. The younger Cuomo then served as secretary of housing and urban development in Clinton’s centrist administration. De Blasio worked for Cuomo at HUD. Then, in 2000, he was Hillary Clinton’s campaign manager when she ran for the Senate.

On Wednesday, Bill Clinton administered the oath of office to de Blasio, who put his hand on a Bible that once belonged to President Franklin D. Roosevelt, one of the mayor’s political heroes. But Clinton’s role was not limited to the purely ceremonial. He also delivered his own remarks before the mayor spoke.

In classic Clinton style, he was gracious toward Bloomberg, who took over the city in the aftermath of the terrorist attacks of Sept. 11, 2001, and had to grapple with the impact of the recession that hit the country in 2008. Clinton thanked Bloomberg for leaving New York stronger than he found it.

But on this day, Clinton, as always, was thinking about tomorrow. He wanted no one listening to conclude that he was there to split the difference between the outgoing and incoming mayors, to offer praise in equal amounts.

And so, after his generous words for Bloomberg, he said, “I have to say this. I strongly endorse Bill de Blasio’s core campaign commitment that we have to have a city of shared opportunity, shared prosperity, shared responsibilities.”

The former president was clearly mindful that the Democratic Party of 2014 and of Barack Obama is not quite the same as the one he led in the 1990s — and that one potential obstacle in the path of Hillary Clinton’s possible presidential ambitions is a primary challenge from the left. His embrace of de Blasio’s message was a deliberate step in the positioning of the Clintons as they look to a possible campaign.

Still, Clinton’s words, though designed to hew closely to de Blasio’s agenda, were his own and carried echoes of the “community, opportunity, responsibility” mantra of his own presidential campaigns.

What he meant in policy terms was not exactly clear. At some point, should she run, Hillary Clinton, who did not speak publicly at the ceremonies, will have to sort all this out. She will be asked to explain more precisely where she stands on issues of income inequality, economic growth, spending, taxes, entitlements and the trade-offs that will face the next president.

Will she be able to be true to the New Democrat ideals that brought her husband to power and also accommodate the energy pulsing through the party’s progressive wing? The answers should begin to come later this year, as she nears a decision about whether to run.