Take a closer look at those heads-and-tails and you'll find some rather disconcerting angles on European history and politics - and a story that goes back to the very first attempt at a European monetary union, 2,500 years ago.
Ironically, given what's been happening in the past few months, that prototype eurozone was masterminded by the Greeks, in ancient Athens, in the middle of the 5th Century BC.
Of course the modern euro designs are a bit of a compromise. From the moment they were invented, coins have always been national and political symbols. The kings of Lydia in modern Turkey, who minted the first metal money about 600BC, didn't do it to facilitate shopping or to heat up the Lydian economy, but to boast of their wealth, power and identity - with the distinctive emblem of a lion stamped on each one.
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- Mary Beard is a Professor of Classics at Cambridge and an author
Some went for simplicity: Ireland features a harp on every single denomination from the two euro right down to the tiny one cent; Estonia, the dullest of the lot, chose a map of Estonia.
But others tried to ring the changes. France, for example, plasters "Liberty, Equality and Fraternity" over some coins, and a strangely angry-looking Marianne, the French Republic's female mascot, on others ("her determined features embody the desire for a sound and lasting Europe" is how the European Central Bank's website attempts to explain this frowning figure).
Italy chose a different work of art for each denomination - with its favourite arena of mass slaughter, the Coliseum, relegated to the five-cent piece. Austria, too, manages a different design for every single coin: from a bust of Mozart to the famous Secession Building in Vienna ("the symbol of the birth of a new age and representing a bridge to a new monetary era", as the ECB again earnestly - and with hindsight rather poignantly - insists).
But it is the Greek euro-coinage that offers the most food for thought. The smallest denominations feature a selection of ships; the 10, 20 and 50 cents each celebrate a modern hero in the struggle for Greek liberation from the Turks; but the one and two euro coins go back in different ways to classical antiquity.
There's no surprise in that. Greek politicians relentlessly harp on about their direct inheritance from the classical world; in fact, I often think that nostalgia is as much a bane on the modern state of Greece as over-spending and poor accounting are.
But in this case, there must be many Greeks who, in retrospect, wonder if the choice of these classical coin designs was quite so clever as it once seemed. Was this pride coming before a fall?
At the centre of the two-euro coin is a bull, and on its back what appears to be a young girl. At first glance, if you didn't know your classical mythology, you might think it was a logo sponsored by the Greens: a symbol of humans and animals living together in harmony.
In fact, it is a rape - Zeus, the king of the gods, snatching Princess Europa from her home city of Tyre, in the Lebanon.
The story was that Zeus had (to put it euphemistically, as most story books do) "fallen in love" with young Europa, and in order to have his way with her, turned himself into a bull and went down to the beach where she was playing with her friends; he nuzzled up to her tamely, licking her hands and encouraging her to stroke him and decorate his horns with flowers.
I've always found it hard to understand quite how the Greek people so easily came to terms with the idea of having a picture of rape jingling around amongst the small change”
But the bottom line was - whatever the pleas in mitigation might be - this was rape.
You can see why the Greeks might have wanted the scene on their euros. Despite the inconvenient fact that Princess Europa came from the Lebanon, her name was eventually given to the continent of Europe. The emblem of the myth on the coin amounted to the claim that without Greece there would have been no Europe - that Greece had invented the continent.
All the same, I've always found it hard to understand quite how the Greek people so easily came to terms with the idea of having a picture of rape jingling around among the small change in their pockets.
Did they not think about the back story to this charming image of girl and bull, or about what was going to happen next? And how does it feel now? With Greek journalists talking luridly about the "rape" of their country and how it has been "shafted", what new meaning does this story of Europa and bull take on?
But Greece's one euro leads in even more intriguing - and ambivalent - directions. Its design is an exact copy of a 5th-Century BC Athenian four-drachma coin, featuring a beady-eyed owl that stares out at you.
This little bird was the symbol of Athena, goddess of wisdom and cunning - and the protector of the city of Athens. Almost all ancient Athenian coins carried this emblem, making the phrase "taking owls to Athens" more or less the equivalent of the modern "coals to Newcastle".
But this is where the first monetary union comes in.
Athens in the 5th Century BC was a democracy (the world's first, so the modern Greeks rather dubiously claim). It was also an exploitative empire, controlling many other states around the Mediterranean.
Some time about 440BC, the Athenians decided to make all these people get rid of their own currency, with their own national emblems, and use Athenian "owls" instead - in fact some of the public notices laying this down, still survive, inscribed on stone around what was the Athenian empire.
The rules, as we can read, were pretty stringent: it wasn't only money that was involved but Athenian weights and measures too (it was the ancient equivalent of imposing grams and metres as well as the euro); and the subject states had to bring their coins to be changed in Athens, with a substantial rake-off for the Athenian treasury. Non-compliance could result in the loss of citizenship; it might even lead to the death penalty.
Modern historians have puzzled endlessly about what was really going on here. Were the Athenians out simply to make a profit? Were the states of the empire actually keen to come into the Athenian-zone? If so, threats of the death penalty seem a bit unnecessary.
And did the legislation actually succeed in eradicating the other currencies and establishing the owls across the empire? We can't be sure.
But on any interpretation, it is hard to resist the conclusion that the Athenian imperialists were using monetary union to display their political muscle - and hard not to imagine that vengeance for that has finally come, 25 centuries later.
But before we get too smug, or laugh at the Greeks for their ill-advised choices of logo - whether rape or owls - we should look at our own.
Last week a foreign student in Cambridge showed me his residence permit for the UK, the bit of plastic, plus biometric chip, that was to be the prototype for the ID cards that everyone was threatened with - but is now issued only to visa holders.
What do you think the symbol in the top left hand corner is? It's the European stars and - the Home Office has confirmed - that very same bull. The female victim has disappeared - it's just the rapist that now guarantees the foreigner a right to live here.