As much as they are about anything, the novels are about subversion. And about how an act of self-preservation becomes an act of subversion. It is about refusing to play the game on someone else's terms--and how an individual struggle can take on political significance. When Katniss refuses--quite literally--to play the game, she unwittingly subverts the authority of the Capitol. Katniss and Peeta work their own brand of subversive magic on their tour, particularly in District 11, and every costume that Cinna creates is a more "Capitol glam" model of subversion. Cinna, frankly, is an amazing character. But Cinna's character in particular contrasts intentional and artistic subversion--priviledged subversion--with artless, spontaneous subversion, which Katniss's particular talent (not fashion design). And for the other players, Katniss is the canvas or clay--a medium for their art.
It's difficult to write about the subversive elements in the abstract, which is, in part, why this post has taken longer to write than the others on Hunger Games. But there's more to the way the novels represent oppression and resistence than a series of identifiable subversive acts. There are the acts of complicity with the oppressive system as well. And again, Cinna is key when, in The Hunger Games, he regards Katniss and says simply, "You must hate us." The Hunger Games illustrates, better than anything I read in graduate school, Center-Periphery theory, and echoes other postcolonial theorists (Homi Bhabha comes to mind) rather nicely. It's sweet, really. And if you missed it, it's central to Catching Fire. Literally.
The dominant image in The Hunger Games and Catching Fire (though not immediately apparent) is a clock--as we learn as the tributes unravel the mystery of the Quarter Quell arena. It could register as clever or simplistic, depending on whether you are predisposed to grant the novel its artistic due. I side with clever--and here's why:
In the arena, each hour represents a punishment that the tributes must endure--or die. The cornucopia--the symbolic (and highly ironic) generosity of the Capitol--is the center, and the point of radiation, and, when the tributes have become too adept at navigating the arena, the point of rotation and disorientation. The Center and the Capitol are linked by the Head Gamemaker's control over when the clock spins to confuse the tributes, who represent the Districts. It doesn't matter that Panem doesn't seem to be arranged in a true circle--or at least, that this reader doesn't have a clear sense of its geography. (I'm sure there are official and unofficial maps online.) The position of the Capitol as the Center, with the Districts occupying the Periphery, is a structure of power rather than geography. The image is also Foucauldian.
In Discipline and Punish, Foucault cites the Panopticon, a prison model developed by utilitarian Jeremy Bentham, as the epitome of the modern means of using discipline as a means of control and coersion to normalize wayward elements within society. The visual image is a circle with a central vantage point from which prisoners might be observed at all times. Here are two images of Bentam's Panopticon, which also bears a resemblance to a certain Flavian Amphitheatre--look! An arena.

Interestingly, in addition to sounding rather like "Pan-American," Panem shares the "pan-" root with "Panopticon" as well. It might be a stretch, and I have a nagging feeling that I'm overlooking the obvious "Panem" reference. But certainly the Capitol also has a panoramic view of the Districts. All related linguistically.
Consider, also, District 13. It doesn't exist. It can't exist. A clock only has 12 hours. So the existence of District 13 spoils the perfection of the clock--and defies the image of the Capitol's control. (Subversion!)
Related to the relationship between the Center and Periphery, on the other hand, is the old adage "Power corrupts." Though I can't find the precise theoretical reference I'm looking for, consider this quote from The Colonizer and the Colonized (1965) by Albert Memmi:
"It is no exaggeration to say that, just as the colonial situation corrupts the European in the colonies, the colonialist is the seed of corruption in the mother country."This theory forms the basis for much postcolonial criticism. And if the relationship between the Capitol and the Districts isn't precisely Colonizer and Colonized, it is as good (bad?) as a colonial relationship. The Districts are, in fact, made "Others" by the rhetoric and practices of the Capitol, which occupies the priviledged "Center" of colonialism--evoking another key postcolonial theorist, Edward Said, of Orientalism fame.
But how does the Center-Periphery power relationship corrupt the Capitol? First, let me be clear that the relationship between the Center and Periphery in postcolonial theory has a lot to do with replication. In the Districts, you don't see little models of the Capitol. The subjugated is not trying to be the oppressor--not really. Except the Career Tributes, perhaps. The Districts don't even have the ability to follow fashion trends. But the power of the Capitol is replicated in the Districts as the people internalize their oppression. The subversive element in Center-Periphery theory is that the oppressed/colonized "Other"--whoever is set apart as "different" in relation to the dominant power--also influences the oppressor. In this case, the corruption isn't to destabilize the identity of the colonizer because the colonized "Other" is encroaching on the oppressor's identity (which is part of the argument in Declan Kiberd's Inventing Ireland as to why the Irish had to be guarded more fiercely than the colonies that were further away, and had a more physically identifiable "Other")--except that Katniss does sort of become "Capitol-ized," if you will. (As have the other tributes.) The citizens of the Capitol begin to feel that she is one of their own, which does destabilize their own identities as it influences their ability to "enjoy" the Quarter Quell. But more clearly, the Capitol--and, by implication, each Capitol citizen--is corrupted by the act of oppressing other people. It makes them monsters, as Cinna recognizes.
The Capitol is the center of corruption. And not only political corruption--all things that are good or pleasurable are found in immoral, ostentatious, and degenerate form, from the purgatives at the banquet to the fashion extremes represented most devastatingly by Tigris in Mockingjay. The closer one gets to the Capitol--Disctricts 1-3--the greater the corruption, with youth training for the slaughter of the Games (and the resulting glory). All of the Districts have internalized their oppression--the "Careers'" Districts have simply internalized them more--proactively, one might say.
Meanwhile, in the Capitol, the people have internalized the decadance of power. Their lives are degraded, and the level of degradation becomes most apparent in their highest form of entertainment--the Games themselves. And ultimately, the Games--also the most visible sign of the Capitol's control over the Districts--become the Capitol's undoing. Turning the means of control back on the oppressor: it's almost a dictionary definition of "subversive."