Here's a good article in Slate's Twins Week by writer Barry Harbaugh about getting a genetic test to see if he and his twin brother Russ, a filmmaker, are identical twins (as they've always believed) or if they might be fraternal. 

The theme of the article is something I wrote about for Taki's Magazine last year: that identical twins are more likely to incorrectly call themselves fraternal than fraternal twins are to incorrectly call themselves identical. What Freud called the narcissism of small differences operates on twins. In 2010, I concluded: "While movie twins look alike but act wildly different, real-life twins often see themselves as less similar, both in looks and personality, than they appear to strangers."

Barry Harbaugh writes:

And yet: We are not strictly identical. We have our petty discordances, which in their accumulation conspire against us. Suspicion is lured by doubt. In a giant senior-class photo of Russell and me that hangs framed in my old bedroom, we might as well be cousins. I am perched at his left shoulder, looking like my head is balancing on a drinking straw, while his own neck threatens to split his collar. We had long ago compared our fingerprints in vain. He was an All-State high-school quarterback (and an All-American in college), while I sat on the bench for a beleaguered basketball team that couldn't win even five games.

Russ was a small college All-American at Wabash. (At first, I guessed that they are related to Jim Harbaugh, a former NFL quarterback now an NFL head coach, whose brother John is also an NFL head coach. But I can't find any proof for that. Still, I wouldn't be surprised if they are all part of an extended family. All these Harbaughs come from the same part of the country — Indiana, Ohio, Michigan — and are all motivated and talented.)

Though we've both escaped the primordial sludge of the Ohio River Valley for New York City, no one confuses us anymore. For two years, I've spent $80 a month on a pharmaceutical that will keep me from going bald. We've noticed that I'm slightly taller. That our noses have a slightly different bent. Our penmanship is at odds and his hair is falling out. ... 

We brought this upon ourselves. Russell and I tried very hard to cultivate individual interests and attitudes. Without surveying the various parenting fashions of twins in history, I might point out the school of thought that demands parents dress their identical children in matching outfits, parting their hair on opposite sides (like a mirror's reflection!), and just as well the school filled with parenting books advising the opposite.
However we came to it—whether through a mother's growing devotion to those books, or some innate desire on our part to complement each other—Russell and I have long attached ourselves to different things, and driven each other crazy with a manic desire to report in detail whatever the other missed. 

This tendency to differentiate can be fairly inevitable, especially among very ambitious twins like the Harbaughs, who are the sons of college professors. If you are named Harbaugh, the top position in sports is quarterback, and only one of you can be the starting quarterback, assuming you don't go to different high schools. If you are named, say, Barber, you can go to the NFL as a cornerback and as a running back and be roughly equally successful. But if you are named Harbaugh, well, there's quarterback, and then there's placekicker, punter, tight end, fullback, center, placekick holder, wedge-breaker-upper, waterboy, and various other football jobs that aren't anywhere near as good as being quarterback. So, it makes sense for one identical twin to break the logjam for both of them by saying, "I don't want to be quarterback. Being quarterback is stupid. I don't care about being quarterback. I want to be [something very opposite of being quarterback, like being a writer.]"

That's fairly important to grasp for thinking about twin studies of nature and nurture that compare identical and fraternal twins. Yes, it is true that other people treat identical twins more identically, on average, than fraternal twins. But identical twins have subtle stresses pushing them apart that fraternal twins are less prone to. If one fraternal twin is built like a quarterback, the odds are that the other one isn't. But if one identical twin is built like a quarterback, then the other one is pretty similar. So, they have to generate their own differentiating forces.

All twins were beheld under the same banner: unusual, unexplained, and undifferentiated.

The confusion was bound with our ignorance of knowing exactly what was happening in utero. It wasn't until the latter decades of the 19th century that embryologists figured out the basic twinning process: Either two sperm fertilize two eggs, or one egg splits in two. (An earlier notion held that twins arose from two sperm that fertilized an ovum in separate places; obviously a red herring.) In 1875, the statistician Francis Galton interviewed 100 pairs of same-age, same-sex siblings, along with their close relatives, and concluded, "Twins is a vague expression." Though not quite a zygotic eureka, he found that extreme similarity among twins was just as common as a moderate resemblance, or hardly any resemblance at all. Even through the embryonic fog, it was clear something elemental divided at least the extremely similar from all the rest. By 1911, the usage of fraternal, as it relates to twins, had entered English usage, and the lexicography of mono- and dizygotic followed five years later. 

Galton's work led to the establishment of the twin method, which proved the foundation for investigations into those dubious sciences called behavioral and eugenic. It also corroborated something that would have been apparent to the era's midwives and cowboys: Not every pair of twins comes into the world trailing the same debris. If you're witness to enough twin births—among humans or cattle—you're likely to notice certain differences from one to the next; that some pairs are born with a single placenta, that others have two placentas fused together, and still more spring from the womb with one placenta each. ...

In 2002, we went off to Wabash College together, a fine all-male school devoted to forging "gentlemen" in upstate Indiana, and one that I stomached for a year before transferring home. The decision was unilateral and it stung. So too: a trip I took that summer to Europe, alone. By sheer coincidence, his football team traveled to Vienna in July, to play an exhibition, and Russell got a tattoo on his left shoulder blade from a man he met in a bar, while I watched. An R whose leg has been cleaved in half so that with a bottom curl it also conveys a B, it proved an unreciprocated mark.

I wouldn't be surprised if there were a big difference between the two twins, one that turns up more often than you'd think among identical twins, one that may have motivated their getting a genetic test to see if they really are identical. (Or I might just be reading slightly more into the various self-dramatizing accounts by the Harbaugh twins than is really there.)

Barry Harbaugh did a lot more research on this question than I did, and came to an even stronger conclusion:

Fraternal twins rarely, if ever, think themselves as alike as two peas. Far more often, misdiagnosis occurs when identical twins think themselves unalike—"of a family likeness only." (In other words, fraternal twins don't question their zygosity; it's the identical ones who get confused). You may have heard that Mary Kate and Ashley Olsen—twins so alike that they have shared professional lives since they were 6 months old, most notably by pretending to be each other without any of their millions fans noticing—have long declared themselves fraternal. What can one say other than: Bahooey. 

As I waited for my own results to come through, I contacted every lab I could find that does this sort of commercial twin testing. I wanted to find stories of identical pairs who had thought themselves fraternal, or of twins that had no idea either way. The search turned up one pair of girls who had grown up with a triplet brother, and couldn't believe the identical result they received: "I never did feel like I was looking at a reflection," one of the sisters wrote. "When the truth finally came out my mom was shocked. She was the mother, how could she get it wrong?" 

But when I asked for help finding a pair that had thought themselves identical, only to discover otherwise—and I shook the bushes for two weeks, badgering labs all over the country—I was met each and every time with silence. Affiliated Genetics, which has been testing for zygosity since 1994, didn't have a single adult twin on record who received a heterozygous result. Not one pair (remotely around my age) had ever tested as fraternal. It seemed nothing short of astonishing. Not one? Where are all the dizygotic wannabes, vainly dressing in matching overalls?