With more than 36,000 cities, towns and villages (all officially called “communes”), France has more municipalities than any other developed country. So nationwide municipal elections, which happen every six years, are a big event. Last Sunday was the first round; most major towns will vote again next Sunday in a second round.

The headline news is that the elections are a setback for Socialist President Francois Hollande and his Prime Minister Jean-Marc Ayrault. The Left appears to have lost ground throughout the country, both to the main opposition party, the centre-right UMP, but also to the far-right National Front. (Official results are here; Le Monde has a more user-friendly version).

That matters on a few levels. First, it’s a psychological blow to a government that’s already going through difficult times, with a sluggish economy and depressingly bad poll ratings. Moreover, French municipalities are powerful institutions, with considerable sharing of personnel between local and national levels. Municipal councils also choose most of the delegates who elect the French Senate, potentially putting at risk the government’s majority when a third of the Senate faces re-election later this year.

From the point of view of national trends, however, it has to be said that the predictive power of local and regional elections is far from perfect. In 2004, for example, the centre-left won a landslide in regional elections, gaining control of 20 of the 22 regions, but it still lost the presidential election three years later. French voters are evidently quite capable of distinguishing between the different levels of government.

On the strength of Sunday’s voting, the centre-right looks like taking control of a number of major towns, including Amiens, Belfort, Caen and Valence. Turnout, however, was a low 61%, so it’s possible that more centre-left voters will be motivated to show up in the second round and tilt the balance a little.

The biggest prize, Paris, is very close, but the Socialists appear to have a slight edge. Analysis of its results is more difficult because, together with Lyon and Marseille but unlike all other municipalities, voting is by ward rather than for the city as a whole.

The media coverage has focused strongly on the swing to the National Front, whose vote in major cities more than doubled from 2008. Its ticket scored a first-round victory in the northern city of Hénin-Beaumont, and leads in several others, including Avignon, Forbach, Fréjus and Perpignan. Nonetheless, it is unlikely to end up with control of more than a handful of councils.

To see why, it’s necessary to understand a bit about how the voting system works (This is for cities and towns with more than 1000 inhabitants, which of course is where almost all the political action is).

Voters vote not for individual candidates but for lists presented by parties or coalitions. To win, a list has to either get an absolute majority in the first round or a plurality in the second round, for which lists that scored less than 10% are eliminated. Lists also have the opportunity to withdraw after the first round and form coalitions to maximise their chances.

The winning list is then guaranteed a majority on the council — basically half of the seats plus its proportional share of the other half. So if three parties scored 50%, 30% and 20% of the second round vote, they would finish respectively with about 75%, 15% and 10% of the seats.

The system therefore gives plenty of scope for deal-making, particularly in the week between the first and second rounds. The Socialist Party will merge its lists with other left groups, including the Greens and the Communist-dominated Left Front, in order to consolidate their joint vote.

The Socialists have offered to do the same with the UMP in cases where that provides the best chance of beating the National Front, as an expression of “republican solidarity”. But the UMP is not willing to go that far. It professes neutrality between the Left and the far Right, although there is clearly a risk that many of its voters will defect to the National Front in the hope of beating the Left.

The UMP, of course, is not alone in facing the dilemma of how to deal with the far Right. It’s become a common theme in the politics of Europe, with responses ranging from the aggressive prosecution of Greece’s Golden Dawn (by a centre-right government) to the way Slovakia’s centre-left reduced the far right to irrelevance by taking them into coalition.

No single strategy works for all circumstances. France’s National Front doesn’t have quite the Nazi-like appearance of Golden Dawn, but it’s toxic enough. The question is whether locking it out of influence altogether is a necessary precaution or whether it just feeds its narrative of victimisation by an elite conspiracy.

But local elections also pose something of a dilemma for the far Right itself. Its totemic issues, most obviously immigration, are national ones, and national posturing by a minor party is a much easier task than actually getting involved in local administration. There will be a tendency for its local representatives to care much more about expressing their anger at foreigners and subversives than about ensuring that local services are run properly.

In turn, that reflects the duality of a nationwide local election. Inevitably it’s about national trends, but it’s also about governing the places where people actually live. Elevating local councils to national prominence may be a mixed blessing.