“The Medici: Portraits and Politics 1512-1570,” an absorbing new exhibition at the Metropolitan Museum of Art, has moments so powerful they sting like the pangs of unrequited love. It helps, of course, that many of the men and women depicted in these 16th-century likenesses were physically beautiful, had exquisite taste in clothing, and enjoyed the means to dress with maximal fashion and luxury. And many of them were depicted when they were young, in the first flush of youth or the vigor of their prime years of power and ambition. But there’s something more to these enigmatic people with power and privilege, and it gets to the heart of this survey of portraiture at a moment of profound political upheaval: They have mastered the erotics of reticence and mystery.

A decade after the Met helped organize an exhibition devoted to Italian Renaissance portraits from the 15th century, the museum has followed with this smaller but generous exhibition focused on the decades just before and after the 16th-century Medici return to power in Florence. The show includes some 90 works, by Raphael, Jacopo Pontormo, Benvenuto Cellini, Agnolo Bronzino and Francesco Salviati, tracing changes in portraiture parallel to a profound political transformation — from Florence as a republic dominated by Medici power brokers to Florence as a hereditary duchy, ruled by Medici autocrats.

There is no clear before-and-after in this, but there are key episodes and an evolution from portraiture as a natural likeness to portraiture as a symbolic, allegorical and literary form. Some of the figures in the early paintings, when the old republic was still in power or a fresh memory, present themselves rather like American politicians did in the early days of the U.S. democratic experiment: soberly dressed in black, with serious expressions, and set against simple backdrops. Showiness and display are not republican virtues.

Later, as the Medici consolidated ducal power under Cosimo I, the dress becomes more sumptuous and little props creep in, especially books, including volumes by Petrarch and other authors who wrote not in Latin, but in the Italian of Tuscany. Details of jewelry, furniture and architectural backdrops take on greater power, situating their subjects within the dynastic and cultural politics of the Medicean regime. Guest curator Carlo Falciani suggests that these signify a move from imitare to ritrarre, or from naturalist depiction to portrayal.

In contemporary terms, we might think of the difference between simply posting our picture online vs. using a picture supplemented by our likes and interests. In 16th-century Florence, however, the supplemental information that creeps into later portraits functions less as psychological data and more as indications of allegiance. It isn’t so much “I like long walks on the beach and reading in the evening,” but rather, “I belong to the tribe that reads Petrarch.”

These telling additions to the paintings can be relatively subtle (a small book clutched close to the heart) and fairly easily deciphered (a few verses, clearly legible, by a well-known poet). But they also can take over the image, as in Bronzino’s gorgeous portrait of the distinguished poetess Laura Battiferri, who holds a large volume of Petrarch’s sonnets that absorbs much of the pictorial energy. And some of them remain mysterious to this day, including the weird concatenation of allegorical references (small renderings of a lion, a naked man, a curiously prominent flower) seen behind the right shoulder of an elegant young man painted by Salviati.

Are these merely puzzles to be decoded to the perennial delight of art historians? The Met exhibition argues that they reveal larger cultural currents. Among them was the decline of Latin and the rise of Italian as a respectable medium for intellectual and literary endeavor. Once Italian became a medium for thinking and feeling, not just a language for daily interaction, a new world opened up, including an increasingly personal intensity of religious expression.

That linguistic shift had political dimensions, too, as Cosimo championed Italian within the larger and relatively fusty framework of Latin as a Pan-European language with an ancient pedigree. The Spanish empire that helped install Cosimo in power was a Latin-infused institution, where mastery of the language marked one as educated and elite. Latin also was a link to the ancient political colossus that still inspired emulation among aspiring autocrats, and in several of the portraits of Cosimo, including two magnificent busts by Cellini, he is represented as a Roman emperor. But while Cosimo deployed Roman imagery when it served his needs, he also supported Italian in part to distinguish Florence within the empire it served.

So, the curious ciphers and clues in these paintings situate their subjects within a complex web of cultural and political change. But the paradox, rather like the banalities of what we call “personal” information today (“I like long walks on the beach”), is that the coded references often feel more like a mask than a revelation. Latin was a tool that increasingly confined its users to seeing the world through the limited prism of Latin literature, allegory and imagery. But Petrarch also was a tool, in which people seemed to see the world through the prism of Petrarchan imagery, especially when it came to the private passions. Reading expands the mind, to be sure, but it also is a good way to hide from the world.

That seems to be what these people are doing, and that hiding is what gives the paintings so much of their power. In a cogent and illuminating catalogue essay, scholar Elizabeth Cropper suggests that the aloofness we detect in these paintings is, in fact, all about “irony and concealment,” as the subjects navigate a world of shifting values and expectations, especially about self-presentation and public image. And if some of them seem a little sad, it may well have to do with a sense of loss, as a republic was subsumed into an empire, as a settled world of Latin thought gave way to an unsettled world of vernacular language. Similar feelings have doubtless been captured in recent photographs, as Americans lament the potential loss of their republic within an epistemological shift away from truth and evidence.

In some cases, concealment operates even on the surface of the image. In Bronzino’s 1546 portrait of Stefano Colonna, the paint is so smooth, the surface so finished, that one realizes this isn’t just an image of a man in armor, but also a metaphor that equates painting with armor, as if to suggest an arrow shot at this panel would bounce off with a sharp clang. In the same artist’s portrait of Battiferri, the poet is as closed to us as her book is ostentatiously open.

This is where the unrequited love comes in. The people in these paintings will simply not tell us who they are, which is why we can’t stop staring at them. In the age they were made, philosophers debated endlessly about the relative merits of painting vs. poetry to recall the presence of an absent thing, especially a loved one. More recently, smart thinkers have abandoned the argument, concluding that neither will ever be adequate to bring us the thing we desire. “Alas, however hard I look, I discover nothing,” the French critic and philosopher Roland Barthes wrote of a cherished image. And as we accept that failure as being inherent to the power of art, we may also draw larger conclusions, about the dispiriting remoteness of love, ideas and even happiness.

The people in these images seem further along than we might have thought when it comes to understanding art as a glorious history of abject failure, analogous to our larger quest to make presence real, and subsume beauty into ourselves. Perhaps their power and privilege helped them get there. It so often does, teaching those who have it only one lesson: There is never enough. We can let another tyrant, not worth a fig compared to the Medici, have the last word.

“There is no happiness in life,” Russian President Vladimir Putin said, quoting Tolstoy, after the recent presidential summit in Geneva. “There is only a mirage on the horizon.”

The Medici: Portraits and Politics, 1512-1570 Saturday through Oct. 11 at the Metropolitan Museum of Art, New York. metmuseum.org.