The gilets jaunes movement is the longest-running protest in France since the second world war. It has lasted for 13 weeks but has mobilised fully for only 13 days – all of them Saturdays.

In the protesters’ own eyes, this is not a protest. It is an insurrection which happens mostly at the weekends. Through a series of Saturday putsches, the yellow vests hope not merely to bring down President Emmanuel Macron but to rip up the constitution of the Fifth Republic and replace representative democracy with popular government. Their support, always limited compared with previous French rebellions, is waning. They will not give up easily or soon. They are mostly peaceful, but acts and words of great violence are committed and spoken.

There is no simple explanation for the gilets jaunes. It is not a monolithic, single-minded movement. It has no leadership structure, no single, accepted programme of demands. That’s what makes it fascinating. And worrying.

In May last year, a young businesswoman of French West Indian origin, Priscillia Ludosky, 31, placed a petition online complaining about the high cost of petrol and diesel in France. Reaction – practically nothing. Ludosky is a calm young woman with dreadlocks who runs her own cosmetics business from home. She is an unlikely pioneer for a populist movement which is sometimes accused of being racist and far right (as parts of it undoubtedly are).

In October she was contacted by Eric Drouet, a 33-year-old lorry driver, who teamed up with her to promote her original petition. Drouet is a car fanatic, a “petrol head”, but also someone with extremist political tendencies. He is now, many weeks later, probably the most influential figure in the gilets jaunes.

Priscillia Ludosky started the movement with her online petition against the rising cost of fuel. Photograph: Eric Feferberg/AFP/Getty Images

There is confusion in France about whether he should be seen as far right or far left. Drouet’s original motivation may well have been his car obsession rather than his politics. Pre-gilets jaunes, he was a loud-mouthed young man more likely to watch Top Gear than to read Das Kapital or Mein Kampf.

By the time he intervened, petrol prices were spiking because of a rise in world oil prices. Ludosky’s original petition exploded online, attracting hundreds of thousands of signatures.

It was Drouet, from Melun in the brie cheese country east of Paris, who thought of the idea of a nationwide protest against fuel taxes on 17 November. Someone else had the brilliant PR idea of dressing everyone up in the yellow hi-vis vests that French motorists must by law carry in their cars. Petrol prices rapidly set alight other grievances in rural and outer suburban France, some concrete, some more existential: a lack of public services, the high cost of living, a new tax on some pensions, the fact that Macron had partially abolished a tax on wealth. No one should underestimate the importance of a decision last July to reduce the two-lane speed limit in France from 90kph to 80kph.

This aroused a long-simmering belief in “peripheral France” that the countryside and outer suburbs are somehow subsidising the insolent success of the cities. Speeding fines, in this rural view, are just another way of taxing ploucs or pecnos – yokels or rednecks. There is also a belief that lower and middle France is taxed unfairly in favour of the rich.

Both beliefs are factually untrue. If anything, the rich subsidise the public services of the poor and lower-middle. The metropolitan areas subsidise peripheral France.

But it is true that energy, life and local sources of wealth have been sucked out of large swathes of France in recent decades – as they have in parts of Britain or the United States. All this adds up to an existential conviction that peripheral France is not only being left behind but mocked and cheated by those who are forging ahead. I believe that this resentment – a sense of being slighted or ignored or despised or abandoned or humiliated – explains the yellow-vest movement more than any particular grievance.

The first day of action on 17 November mobilised 283,000 people across France. This figure has never been reached since. It was impressive but fell well below the numbers that have turned out for other social protests in the last 20 years. Its significance was that it took place not just in cities but also in small towns and on almost every local roundabout. Like the Tour de France, it was a national event which came “down your way”.

People who had previously felt powerless felt abruptly powerful because they could hold up cars and lorries at a roundabout. People who had been invisible became highly visible in their hi-vis vests.

Here is our second paradox. The gilets jaunes are an internet-spawned and driven movement. They could not have existed before Facebook existed. But their attraction is that they are also something tangible, a social club as well as a social protest.

The gilets jaunes were brought to a white-heat pitch of anger by the halls of facing mirrors of “anger groups” online. But they also allowed people to escape from the isolation of their terminals or smartphones to come together and do things.

The 17 November protests – Act One – were mostly peaceful. There was little violence, except a few scuffles with police in Paris and with motorists on blocked roundabouts. The government stood back and allowed the protests to happen, even though the gilets jaunes deliberately defied the law which demands advance permissions for large gatherings.

It is important to remember this when gilets jaunes and Russian news sites allege that Macron has tried to “suppress” the movement.

On 24 November, Act Two of the movement, there was some violence in Paris. Police responded robustly to being attacked by a militant fringe of protesters on the Champs-Élysées. There were several serious injuries from police weapons. This was treated on GJ sites as deliberate police violence and repression, stoking tempers for the next weekend.

On 1 December, Act Three, I was out on streets of Paris from early morning. I happen to live near the Arc de Triomphe, and when I left my building at 9am I walked out into the middle of a street battle. A number of yellow vests – not just hard left and right militants but rural or provincial disaffected men and women in their 20s, 30s and 40s – attacked police from early morning.

Flames at the Arc de Triomphe in Paris in November. Photograph: Bertrand Guay/AFP/Getty Images

The Arc de Triomphe was tagged with graffiti. Police were pelted with acid, paint, iron bolts, stones and bottles. Buildings around the Étoile were set briefly alight. A mob surged down the Avenue Kléber, overturning and burning cars and smashing bank, shop and restaurant windows.

Before the next Saturday – Act Four on 8 December – there were dark reports that there would be an insurrection, using live weapons and explosives. Macron, shaken, visited the Élysée nuclear bunker to make sure there was a place that he could take refuge. In the event, 8 December was again a violent day – not just in Paris but in several cities. It was not the feared revolution. The violence this time was mostly provoked by militia of the hard right and hard left. Some “ordinary” GJs waded in. Police were more aggressive in their response.

Since then there have been nine further Acts or Saturday putsches, some more violent than others, some better supported than others. Violence has switched from Paris for the most part to provincial cities, especially Toulouse, Bordeaux, Caen, Rouen and Rennes. The numbers taking part have fallen to around 60,000.

The roundabout rebellion is almost over. Support in rural areas is slackening. In early December my own unscientific poll of yellow vests placed on car dashboards in rural Normandy was 40%. It’s now 10% to 14%.

There has been no systematic attempt to repress protest. But the police’s so-called defensive weapons – rubber bullets and stun grenades – have become a serious problem that the government has been too slow to recognise. Seventeen eyes and four hands have been lost so far. That is unacceptable, whoever is initially responsible for the violence.

Is this just the French being the French? Not really. It’s true that protest goes to the street in France more rapidly than in almost any other democratic country. There is a default position in the country that social demands will not be taken seriously without street demos, and demos will not be taken seriously without a dose of violence.

A statue of Marianne, a symbol of France, inside the Arc de Triomphe was damaged by rioters on 1 December. Photograph: Etienne Laurent/EPA

But there is also something very un-French about the gilets jaunes movement. It has, first of all, broken all the unspoken and spoken rules of French manifestations. Protests are usually choreographed within certain limits, with agreed venues and routes and marshals. Riots happen but they are predictable riots.

The gilets jaunes refused from the beginning to be bound by any of these rules, although this is now changing. The movement, supposedly peaceful, allowed its own violent fringe as well as opportunist, urban, state-hating militia of hard right and hard left to take over.

The violence is not limited to the Saturdays-only protests in French cities. There have been scores of arson and other attacks on the offices of MPs who support Macron. There have been attacks on motorways toll-booths, newspapers and radio stations. A starred restaurant whose boss criticised the gilets jaunes was burned. Town halls, prefectures and other public buildings have been vandalised. The gilets jaunes are also un-French – or at least atypical – because they have not come from the usual or expected sources of protest: unions, farmers, students, the multi-racial banlieues. They come from a section of French society that is usually invisible, permanently grumpy but little engaged in politics.

I mean rural, non-farming France, small town France, outer suburban France. I mean many people who have not voted for years, as well as many who have voted hard right or hard left.

Gilets jaunes do include people at the bottom of society, the unemployed, the marginalised, but more typically they are people who are above the bottom but think that they should be doing better. They are people with low-paid jobs or pensioners or artisans or small business people or technicians or people at the lower level of the caring professions.

President Emmanuel Macron speaks during a special address to the nation on 10 December. Photograph: Ludovic Marin/AFP/Getty Images

This mixture of supporters explains in part the heterogenous character and demands of the movement, which point both left and right, to higher welfare payments and pensions but also to lower taxes, to less state and more state. There is no coherent ideology, even a refusal of ideology.

Some see this as a camouflage for a movement whose heart beats on the far right. But many of the usual far-right hot topics – migration, Islam, abortion, gay marriage, Europe – are not the first things that you hear on the lips of gilets jaunes.

What are the sources of grievance and how justified are they? Here it is difficult to prise apart French and un-French elements.

The words you hear most often are mépris and ras le bol. Mépris means contempt. Gilets jaunes are convinced or have been convinced that the little or middling people like them are held in contempt by the trendy, rich, globally-oriented people of successful metropolitan France.

That’s why, I think, Macron has sparked such anger and hatred: not so much for what he has done in the past 20 months but for what he represents and some of the things he has said. He typifies the kind of person that GJs didn’t like even before they were GJs or before there was Macron.

He is the embodiment of the rich, clever, self-replicating people from the governing classes who’ve been to the finishing schools of the governing elite and think they know everything.

Eric Drouet had the idea of calling national protests and the movement exploded into action. Photograph: Zakaria Abdelkafi/AFP/Getty Images

And so to the second word or phrase you hear: ras le bol, which means sick to the teeth. It conveys a sense of having been pushed to the breaking point. In my experience, French people almost always have a sense of ras le bol about something or other. But the white-hot anger of the GJ movement is something new and different. It can only be explained, I think, by the silos of facing mirrors of social media sites, which fill heads with exaggerated versions of real grievances blended with complete bullshit.

“France has been sold to the UN”; “Alsace has been given back to Germany”; “Politicians are living high on the hog – if we sweep them away, we can have lower taxes and higher welfare payments”; “Brigitte Macron, as first lady, earns €550,000 a year from the state”. Actually she earns, nothing, zero, rien.

This nonsense is always accompanied, as it often is in the United States and Britain, by a message telling you that the mainstream media is lying. Thus suspicion of the media, already high, is cranked to the point where journalists, newspapers and radio stations have become targets for GJ violence.

The gilets jaunes rapidly moved from a protest movement with specific demands to a revolutionary movement of unlikely revolutionaries. The view in much of the foreign media is that these are anti-Macron protests. And they are. But they go way beyond him and call for the sluicing out of the entire political class, left and right, even hard left and hard right, to be replaced by a bottom-up, direct democracy of permanent popular decisions by referendums.

Thus we have a revolution which did not just start on the internet and Facebook but wants to use the internet to impose a new form of government and yet is at the same time anti-globalist and nostalgic for a simpler, more traditional kind of France. Yet another paradox.

Riot police stand in position during clashes with demonstrators in Paris, on 1 December. Photograph: Alain Jocard/AFP/Getty Images

The movement is now splitting. Some gilets jaunes have decided to enter the mainstream political system to try to reform it or destroy it from the inside. Three rival GJ lists, broadly left, right and centrist, are being prepared for the European elections in May.

An extraordinary diplomatic row exploded last week when the Italian deputy prime minister, Luigi Di Maio, leader of the populist Five Star Movement, arrived unannounced in the south Paris suburbs for a photo opportunity with one of these electoral lists.

France withdrew its ambassador from Rome in protest. It emerged that Di Maio had been invited by Christophe Chalençon, a gilets jaunes figure with far-right connections, not by the leader of the list, Ingrid Levavasseur. She described both Di Maio and Chalençon as “sharks”.

All of this may help Macron. He is already recovering strongly in the opinion polls as well-heeled, urban France turns to him to save them from yellow chaos. He has made €10bn of concessions to motorists and the low-paid. He has started a “great national debate” in hundreds of village and town meetings.

Even one yellow vest list in the European elections will take votes from Marine Le Pen’s Rassemblement National and Jean-Luc Mélenchon’s far-left La France Insoumise. For Macron’s party to top the European poll in France would be a “mid-term” triumph.

There is talk of Emmanuel Macron calling a referendum on the same day as the EU poll on 26May to take further wind from the sails of the gilets jaunes. This is probably a silly idea. Senior ministers are campaigning against it.

If the referendum asks marginal questions – ie should there be fewer French MPs? – it will seem like a betrayal of the promise of Macron’s “great debate”. If it asks explosive questions – should Macron resign? Should he push ahead with his state-shrinking reform programme? – the results could be catastrophic for him.

LeVavasseur, an auxiliary nurse from Normandy, is one of the more thoughtful gilets jaunes (and therefore detested by many others). She is leading one of the GJ lists in the European elections. Asked to sum up yellow-vest demands in a single sentence, she said: “That we should feel recognised and valued.”

The anger of the gilets jaunes is understandable. What is scary is their fury. Where does that come from? What explains the radicalisation of home carers and garage mechanics in small towns who don’t understand the basics of how their own country’s political and tax system works but want to tear them both down?

That fury has many sources but its depth and strength can only be explained by the compound or viral effect of social media. The gilets jaunes are a French phenomenon and a French crisis.

They also point to a wider, existential crisis for 21st-century democracies.

This article is adapted from a talk given to a conference of the Brussels European Employee Relations Group