The European media are full of angst this morning over the results of elections to the European Parliament, which concluded on Sunday. The elections are being spun as a big victory for extremists and Eurosceptics, throwing the direction of the European Union into doubt. Extremists topped the polls in Britain, France, Greece and Denmark, and scored major gains elsewhere.

There’s certainly some basis for concern, but things are not as dire — or as exciting — as you might think. To understand why, you first need to know a few things about the European Parliament.

The first thing is that the Parliament, although it is gradually becoming more assertive, is not as powerful as a regular national parliament. It has no power to initiate legislation, and although the European Commission — the EU’s executive — is nominally responsible to it, it is mostly drawn from outside and does not rest on party support the way a normal parliamentary government does.

The second thing (for which the first thing is no doubt largely responsible) is that voter interest is very low, and voter focus on EU issues is lower still. Until now, turnout had declined at every election; this year recorded the tiniest of upticks, from 43.0% to 43.1%.

Many of those who do vote clearly treat it like a giant byelection, taking the opportunity to show their dissatisfaction not just with the EU but with their own national governments.

The third thing to know is that because it draws on 28 separate party systems from the member states, the party composition of the Parliament is complex. I had a go at explaining it in a blog post on Sunday, but basically there are party groups for five distinct ideological currents: far Left, centre-Left, liberal/centrist, centre-Right and Greens/regionalist. There’s also a breakaway, European Conservatives and Reformists (ECR), from the main centre-Right group, driven mostly by the British Conservative Party.

That accounts for a total of 701 out of the 765 members of the outgoing Parliament (the new one will have 751). The remaining 64 can almost all be loosely described as “far Right”; 31 of them are in a Eurosceptic party group called Europe of Freedom and Democracy (EFD), of which the biggest component is the UK Independence Party, but the others — including the real extremists, such as those from the British National Party, the French National Front and Hungary’s Jobbik — sit as non-aligned members, or non-inscrits.

With that background, we can look at the weekend’s results. (I’m using numbers from the official EU website; the BBC has a slightly different set, and Wikipedia’s are a little different again).

The mainstream parties have suffered losses but still command the overwhelming majority of seats. The centre-Right remains the biggest group with 213 seats (down 60), or 259 if you add in ECR. The centre-Left has 190 (down 6) and the liberals 64 (down 19). The Greens are also down slightly from 57 to 53.

The remaining 185 are mostly extremists and Eurosceptics, but not all extremists are Eurosceptics, and not all Eurosceptics are extremists.

The far Left has gained ground, going from 35 seats to 42 — mostly due to Syriza, which won a plurality in Greece and took six seats. It will probably also be joined by the new Spanish left-wing party Podemos, which won 8% of the vote and five seats.

The Left, however, although hostile to current EU policies, is not opposed to the EU in principle. This is not the 1930s; there is no prospect of far Left and far Right teaming up to destroy democracy.

Other new parties are Eurosceptic without otherwise being extremist: ANO in the Czech Republic, which won four seats: Alliance for Germany, with seven; Poland’s Congress of the New Right, with four. There is almost nothing in common between them and the neo-Nazi groups like Jobbik (which held its three seats) or Greece’s Golden Dawn (also with three seats).

Somewhere in between are the larger far-Right groups: UKIP, the National Front, the Danish People’s Party, Italy’s Northern League, and Geert Wilders’ Party for Freedom in the Netherlands. Exactly how they will organise themselves in the new Parliament remains to be seen. Wilders and Marine Le Pen have planned a new far-Right Eurosceptic group, but it’s not clear how many will join; UKIP has already made it clear it is not interested.

And not all of the far Right did well. Wilders could only manage third place in the Netherlands (well short of expectations), the Northern League’s vote was also down sharply, and Golden Dawn’s 9.4% of the Greek vote was much less than some polls had been saying.

The British and French results were certainly shocking, but they have more to do with domestic politics in those countries than with a continent-wide shift towards fascism. Other big countries stayed with the mainstream: the centre-Left made gains in Germany and Italy (where Beppe Grillo’s 5 Star Movement finished a distant second), and the far Right in Romania was wiped out.

It would be idle to deny that the EU has some major problems, and if these results force the established politicians to take those problems more seriously — as promised yesterday by French president Francois Hollande — it will be a good thing.

But although the barbarians may be restless, they are not yet climbing over the walls.