It’s either part of his peculiar charm or proof of its absence, depending on your feelings about him, which are surely fixed by now. But Mayor Michael Bloomberg has a way of speaking so gruffly that he causes stirs he can’t have meant to. He doesn’t exactly put his foot in his mouth, not all the way. Just the big toe and maybe the index one, too.

He did it again in an interview with New York magazine’s Chris Smith that was published over the weekend and that belongs to a quickly expanding body of Bloomberg exit literature. I guess when you’ve been mayor as long and large as he has, you get much more than the standard sayonara: you get a laborious countdown, serial autopsies, a clutch of would-be successors appraising you with the kind of warmth accorded the Wicked Witch of the West. Ding-dong. Bill de Blasio is here to wipe clean the civic memory of you.

It was de Blasio who set Bloomberg off. Rather, it was de Blasio’s resonant tale of two New Yorks, the wealthy one that Bloomberg is accused of coddling and the less wealthy one that he supposedly showed the back of his hand.

Bloomberg told New York magazine that de Blasio was running a “class-warfare and racist” campaign. The “racist” part was clumsy overreach. He instantly had to walk back and explain what he meant, which was that de Blasio was cynically showcasing his African-American wife and their kids for liberal cred and minority votes.

But is it any wonder that de Blasio has gotten under the mayor’s skin? That the whole narrative of the Democratic primary has? It’s a narrative of either-or, of winner-loser, of one group’s blessings explaining and in some ways causing another group’s deprivations.

The truth is less callous and more complicated than that.

It is indeed the case that income inequality in New York City has worsened during the Bloomberg years, to an extent that’s morally unacceptable and perhaps socially untenable. But it’s also the case that the situation reflects a national trend. It isn’t principally Bloomberg’s doing, and it surely wasn’t his intent.

To look at his signature initiatives and see only an upper-class agenda is selective, reductive, lazy. And it assumes, in an odd and even condescending fashion, that certain improvements in New York life aren’t appreciable across the whole spectrum of income.

Bloomberg’s remarkable greening of the city, for example, hasn’t merely gilded the scenery for bigwigs zooming past it in their black Escalades or for littler wigs pedaling through it on their bikes. It has given all New Yorkers places to unwind, chances to breathe. I’m frequently in Brooklyn Bridge Park, a splendid oasis, and can tell you that it’s not just a refuge for the affluent denizens of the nearby brownstones in Brooklyn Heights. The entire borough in its remarkable diversity is there. That’s a big part of the park’s glory.

By making smoking more difficult and expensive, Bloomberg wasn’t simply improving the air in Midtown Manhattan boardrooms. He was trying to muscle New Yorkers of all stripes away from an injurious and sometimes deadly habit. While his public-health efforts may have struck some of his haters as elitist and paternalistic, their potential and actual beneficiaries include the city’s less fortunate residents, whose obesity rates, for example, surpass those of the affluent.

Bloomberg didn’t get public schools nearly to the level that we need them to reach. And there was that whole Cathie Black fiasco. But with the help of her predecessor, Joel Klein, he seeded New York with scores of new charter schools, many of them excellent, providing tens of thousands of disadvantaged children with a caliber of education and a kind of hope they didn’t have before.

The drop in crime under his watch has been remarkable — and citywide, not just in neighborhoods with luxury high-rises. Other initiatives, like expanded tax credits, were aimed at poorer New Yorkers. Were they enough? No, but municipal government has finite resources.

Has Bloomberg worshiped at the altar of Wall Street? Yes. But then so, too, have many other politicians, from Chuck Schumer through Cory Booker, who aren’t tagged with the lack of caring that is hung on the mayor, who’s all too easy to caricature and to scapegoat: because he’s a gazillionaire; because he has tapped that wealth to shore up support; because he has that toe-in-mouth disease, and seems at times to shrug rather than rail at how precious and exclusive so much of Manhattan and Brooklyn have become.

How much of a difference would railing make? We’ll probably find out, whether de Blasio or some other candidate prevails. But while more passion about the gap between rich and poor is a virtuous thing, making Bloomberg the heavy isn’t.