More than 400 people were arrested across France on Saturday after protests ostensibly about high school students' grievances collided with the time-honored French tradition of taking to the streets to express displeasure with, well, everything. Police fired tear gas as demonstrations were boosted by other groups angry over high prices and the fraying of a social welfare system - because nothing says "solidarity" like a nationwide coalition of the aggrieved.

Hundreds of thousands of people took to the streets, and tensions erupted on the fringes of the nationwide demonstrations over absent teachers, decrepit school buildings and police violence. Interior Minister Laurent Nuñez said late Saturday that there had been 400 arrests, most of them adults not affiliated with the high school protests. He blamed casseurs - "opportunistic delinquents who had seen a chance to pillage and loot … we will find them and deal with them firmly" - for hijacking the official demonstration, a phrase that has never once preceded a calm evening.

Nuñez said the state was not planning to declare a state of emergency but added: "I very strongly condemn the violence we have seen." The government's official position, then, is strong condemnation without the paperwork.

In Marseille, protesters stormed the town hall, ramming the door with security barriers they had removed from the street. They briefly entered the municipal building, where one protester spray-painted an "A" for anarchy on a wall before police removed them. It was, at least, a concise statement of intent.

Around Place de la République in central Paris, a peaceful march turned violent, with police firing tear gas at black-clad and masked rioters who threw fireworks and paving stones, attacked a bank and looted a sports shop. "We're taking the Bastille!" one demonstrator yelled - a line that would have been more stirring if the target had not been a retail outlet.

Others set wheelie bins on fire and threw firecrackers inside, or pushed over glass recycling bins and picked up the bottles to use as missiles. A lorry was set on fire and flames spread to a nearby building. Along Boulevard Magenta, one of the main south-north axes north of Place de la République running through a mostly working-class area, dozens of shops were broken into and looted. Residents in surrounding streets were forced to close windows and doors as huge clouds of tear gas rose to roof level. French authorities detained dozens of protesters across the country, with more arrests expected.

Saturday's mobilisation followed calls on social media to revive the gilets jaunes (yellow vests) movement, which emerged in 2018 against social injustice. The protests began last month at schools in the poor northern suburbs of Paris, where anger over teacher absences and shabby facilities quickly morphed into a nationwide movement against inequality, weak public spending and falling standards. Saturday's protests called for teachers and students, labourers, pensioners and farmers to join and widen the movement - a broad church with a very loud choir.

French authorities said 140 protests involving 32,800 people had been recorded outside Paris by midday, blaming the unrest and violence on what they said was the "ultra-left." The CGT, one of France's biggest labour unions, backed the protests, calling for "higher wages and pensions, lower prices and the protection of public services that meet people's needs." About 45% of French people are worried about the rising cost of living and inflation, according to a 2,000-person survey conducted in October - a statistic that explains rather a lot.

Protesters in hi-vis vests blocked streets, an echo of the unrest that dogged President Emmanuel Macron's previous term, and in Marseille rubbish bins were set alight. Both the far left and far right sought to capitalise on the movement, with the left positioning itself on the side of the protesters and the right backing the police. Jean-Luc Mélenchon, leader of the far-left France Unbowed party, marched in a demonstration on Tuesday and has expressed support for student protesters.

A large police presence was visible in central Paris, especially near landmarks such as the Avenue des Champs-Élysées and the Arc de Triomphe, which were excluded from the routes of the two official demonstrations. About 100 people walked towards police at the beginning of the protest yelling, "all cops are bastards!" - a sentiment that rarely appears on tourism posters.

"We are fighting because the cost of living is too harsh, because the price of fuel is too high, because high school pupils are being violently attacked and hurt all over the country just for trying to protest and ask for more teachers and decent learning conditions," said a 19-year-old protester in Paris who gave her name as Lola.

Macron is entering the final stretch of his presidency, leaving behind a polarised electorate and a badly fragmented political landscape. On Friday, he appealed for calm and respect for people and property as officials warned of possible violence. "There is a constitutional right to protest and to express oneself, and I have always been vigilant and attentive that that right is fully respected," he said. "But this right comes with the need to respect. Respect for people, respect for property, and therefore a commitment not to insult or destroy." He denounced as "unacceptable" the potential presence of people "solely to cause damage and create unrest." The president's message, in summary: please riot responsibly.

Nuñez said hundreds of protest actions and more than 80 marches were expected nationwide, including spontaneous gatherings, street demonstrations and blockades. Police banned some other proposed gatherings in central Paris. Many Métro stations were closed on Saturday at the police's request, including those around the Champs-Élysées and near the presidential palace, parliament and government offices.

In recent weeks, there have been thousands of arrests, hundreds of injuries, and schools set on fire as the new school year descended into nationwide student protests that are influencing the country's presidential election campaign and drawing international attention. The police watchdog is investigating at least two dozen allegations of police violence, while the French prime minister said more than 900 police officers had been injured. Among protesters, particularly gruesome wounds suffered by some teenagers have been blamed on police riot-control weapons. Nuñez suspended the use of a type of police stun grenade against young demonstrators after a 15-year-old lost a hand, but said they could be used on Saturday - a distinction that will doubtless comfort anyone keeping count.

On Tuesday, climate and social justice activist Greta Thunberg joined students at a school in the Paris suburb of Saint-Denis, proving that the movement's ability to attract international attention remains intact even as the domestic political landscape frays.