* Protest in Rome turns violent, six arrests
* Protests organised by hard-left parties
* In Paris, 25,000 turn out, police say
* France, Italy both trying to cut debt, boost growth
The protest in Rome turned violent when a large splinter group - many wearing masks and helmets - threw rocks, eggs, firecrackers and oranges at riot police in front of the industry ministry.
Riot police with batons charged the group, with protesters fighting back with rocks and firecrackers. One man lost a hand when a firecracker exploded before he could throw it.
There were dozens of lighter injuries among police and protesters, and at least six arrests, police said.
In Paris, protestors marched from the Place de la Republique, some carrying banners attacking President Francois Hollande with slogans such as "Hollande, that's enough" and "When you are leftist you support employees."
French police said that about 25,000 joined the protest, which came after new Prime Minister Manuel Valls unveiled planned tax and spending cuts on Tuesday, vowing to bring down France's public deficit and following on the heels of pro-business reforms announced earlier this year by Hollande.
"This is the first demonstration of the left-wing opposition against the government," Olivie Besancenot, spokesman of the New Anti-Capitalism Party told i<Tele TV channel.
The turnout, however, was well short of demonstrations in Paris last year against same-sex marriage that drew hundreds of thousands. The French Communist Party, on its Twitter account, estimated Saturday's turnout at 100,000.
The protest in Rome was smaller, drawing several thousand, according to witnesses. They called for more affordable housing and took aim at 39-year-old Prime Minister Matteo Renzi and his plans to reform labour rules to make it easier for companies to hire and fire employees.
"The problem with the Renzi government is that since it took power, even though he is supposedly of the left his policies are of the right," said Federico Bicerni, 23, a graduate from Modena with a temporary work contract who is also the youth head of the Italian Marxist Leninist Party.
"They are reducing democracy. Renzi's labour reforms will worsen the situation for workers without job security, hitting young people when they are already struggling. The rage of the people in the squares today is justified," he said.
Renzi, who took power in February, is seeking to make sweeping reforms, including tax cuts, to revive Italy's ailing economy where youth unemployment has risen to well over 40 percent.
French President Hollande is struggling to turn around a weak French economy and hard-left parties have urged him to abandon his business friendly reforms and public sector deficit targets set by the European Union. They say the government's plans are hurting growth and are not helping bring down unemployment, which is above 10 percent.
Hollande announced a so-called "responsibility pact" earlier this year, which aims to boost the profitability of French companies by cutting what they pay in social charges by some 30 billion euros.
"A 30 billion gift to big business is something monstrous in a period of austerity we live in," Left Party leader Jean-Luc Melenchon told RTL radio on Friday ahead of Saturday's march in Paris.
Valls was appointed prime minister in a reshuffle this month after Hollande's Socialist Party suffered a drubbing in recent local elections where the far-right National Front made strong gains. Leftist allies grumble that Valls is too centrist.
Polls show that the ruling Socialist party will finish third in next month's European elections, behind the conservative UMP and the National Front.
"Fiscal discipline and austerity, imposed by the European Commission, Francois Hollande and the government to satisfy financial markets feeds an unheathy climate," union representatives and politicians said in a letter calling on people to join the march, published on Wednesday on the website of L'Humanite newspaper, Humanite.fr.
Paris is under EU disciplinary action for running too high a bugdet gap and has already been granted a two-year delay to reduce it to within the EU limit of 3 percent of gross domestic product by the end of 2015.