For the second time this week, Mayor Rahm Emanuel on Wednesday did something he seldom, if ever, does: make a concession to the City Council on a major issue pivotal to city finances.

At the request of the anti-Emanuel Progressive Caucus, Emanuel shrunk his massive borrowing plan — by another $200 million — to ease concerns about lucrative swap termination fees paid to major banks.

Chicago has already paid $250 million in similar penalties over the last five years. The $200 million in water revenue bonds, dropped for the time being, were billed as the last “variable-rate conversion” involved in terminating those complex deals dating back to the tenure of former Mayor Richard M. Daley. It would have paid the banks at least $100 million more.

“Welcome to the new City Council,” said a triumphant Ald. John Arena (45th), who voted with the majority in a 40-to-4 vote in favor of the revised, $2.35 billion borrowing.

Earlier this week, Emanuel agreed to cut in half his plan to issue $1.25 billion in general obligation bonds backed by property taxes under pressure from aldermen unwilling to give the mayor a virtual blank check. They demanded and will apparently get a list of projects to be bankrolled, more information on interest rates and quarterly reports on city borrowing.

On Wednesday, the mayor blinked again. It was yet another sign that aldermen emboldened by a once-powerful but now wounded Emanuel are no longer willing to give the mayor the benefit of the doubt.

Emanuel took the declaration of independence in stride.

“Before, you all said that they were too much of a rubber stamp and that we should have them express themselves. If you let them work through issues — whether it’s inspector general or bond stuff — then you say it’s a reflection that you don’t have political leadership,” the mayor said.

“You know that phrase which is told by rabbi? Kid comes home. His mother — since it’s a rabbi, you should assume it’s a Jewish mother — gives him a green sweater and a blue sweater. He goes upstairs, puts on the blue sweater and she goes, ‘What’s wrong with the green sweater?’”

In other words, you can’t win.

“The City Council is going to play an important role in being a partner and moving the city forward and I think they’re going to continue to do that,” he said.

Emanuel’s handling of the Laquan McDonald shooting video has turned a once powerful and feared politician who was a major player on the national political stage into a shadow of his former self.

In his struggle to regain shattered public trust and fend off demands for his resignation, Emanuel has repeatedly reversed field after his political instincts betrayed him.

He has fired a police superintendent he promised to keep, welcomed a federal civil rights investigation he once called “misguided” and ordered a third-party review of a Law Department he initially claimed could not possibly be part of the “code of silence” he acknowledges exists in the Chicago Police Department.

All of those moves have made him look desperate and weak and emboldened aldermen to push back to save their own political necks.

Like Emanuel, they were harshly criticized for signing off on a $5 million settlement to McDonald’s family one week after the April 7 runoff — even before a lawsuit had been filed — without asking tough enough questions and seeing the incendiary video.

The retreat over borrowing on Wednesday was only the latest example of that new political reality.

But Emanuel made no apologies for shifting gears.

“Part of being in public life is, if you don’t get it right [the first time], be honest and get it right. And not being scared to say that,”  he said.

“And not just get it right, but get it right in the goal of greater transparency and accountability. And if we didn’t do it, right being honest. There’s an important policy with yourself because, if you’re going to fix something, you’ve got to be honest. And that’s what we did.”

As for the penalties tied to terminating those complex swap agreements, Emanuel said it’s all part of his debt restructuring plan to move away from risky financial practices that Daley had used to “mask” the true cost of city government.

“These are about undoing past financial engineering called swaps. It’s consistent with what people in the financial community and the rating agencies want to see about creating greater stability and more certainty in the city of Chicago. And this is all an effort to unwind things we inherited,” he said.

Arena and Ald. Scott Waguespack (32nd) voted with the majority because the swap termination agreements they have railed against were dropped. Both men want the city to use the delay to explore all options to bring those penalties down.

“They could easily sue. . . . There’s opportunity there to do that. They just haven’t taken those steps. Other alternatives might be going back to the table and trying to negotiate this, instead of just saying, `Here’s $100 million, which we’ll turn into $200 million by the end of the term with the interest,’ ” Waguespack said.

Earlier this week, Budget Director Alex Holt said the swap contracts were entered into “knowingly” by Daley administration officials who “understood the risks.” And a city attorney disclosed that the Law Department examined the possibility of suing the banks and found there was “no legal basis under which securities law or state law” would allow such a challenge.