This past summer, Edward I. Koch, a Democrat, made headlines by noisily endorsing Republican Bob Turner in a special election to fill the congressional seat of disgraced Tweeter Anthony Weiner. The former mayor explained that he’d decided to rally Jewish voters in Brooklyn and Queens to chastise President Obama for his Israel policy. Koch’s outsize role in Turner’s surprise victory made for big political news and led to speculation that Obama could be facing trouble in his reelection bid.

The episode was unusual, but unusual has always been de rigueur for Koch, a three-term mayor and a constant post-mayoral presence in New York City. For one thing, Koch has long described himself as a “liberal with sanity” in a city that consistently tilts left. Further, Koch’s...