By Stephen Mulvey BBC News Online Europe Editor in Paris

It is the second biggest sensation so far of the French election - first the shock success of Jean-Marie Le Pen on 21 April, now the outpouring of more than a million people on to the streets for anti-National Front May Day demonstrations.

The main French newspapers devote several pages to it, and to the much smaller demonstration by Mr Le Pen and his supporters in Paris.

Protesters take Liberation's anti Le Pen message

The left-wing Liberation has on its front page the simple headline "No means No" and suggests inside that it's not since the days of revolutions that so many have protested at once.

Others hark back to 1968, or to the end of World War II.

But precisely what significance of this protest is, is undecided and disputed.

It was predominantly the left that took to the streets - as one conservative deputy put it, marches are not part of the culture of the right.

Huge discrepancy

But does that mean that the left is beginning the process of renewal after its shattering defeat in the first round of the presidential election?

Liberation draws attention to a huge discrepancy in numbers - 1.5 million May Day marchers, compared with the 200,000 votes by which Mr Le Pen pipped the Socialist Lionel Jospin to the second round of the election, and the 20,000 who demonstrated on behalf of the National Front on Wednesday morning.

Unionists, communists, anarchists, greens, anti-racists and gay rights joined the demo says Le Figaro

It quotes pundits questioning the legitimacy of a conservative president elected - as Jacques Chirac will be on Sunday - as a unity candidate, thanks as much to the left and extreme left as his own natural supporters.

And it draws attention once again - quoting Socialist leader Francois Hollande - to the fact that less than 20% of French people voted for Mr Chirac in the first round.

But the right-wing Le Figaro has a rather different take on the situation.

It refers to the "flowering of heterogeneous slogans and symbols" on parade at the big Paris demonstration as unionists, communists, anarchists, greens, anti-racists and gay rights groups marched past the Bastille from the Place de la Republic to the Place de la Nation.

Anti-fascist principles

A columnist in the same paper mocks the unfocused anti-fascist rhetoric of many of the protesters, saying: "How fine it is to march against phantoms! How fine it is to teach your grandfather a lesson: 6 February 1934, Mussolini, Hitler, Petain, we would not have let it happen."

The philosopher Andre Glucksmann comes at the same point from a different angle in the centre-left Le Monde.

He argues that it is useless to put on display one's "splendid and immaculate anti-fascist principles" if you dodge the brutal question, to vote or not to vote for Mr Chirac?

Mr Jospin in particular has yet to respond to requests to call clearly on people to vote for Mr Chirac, "to save France" as the 1968 veteran Daniel Cohn-Bendit puts it.

Despite the May Day sensation, it is very clear that the Socialist Party has a huge amount of work to do before the June parliamentary election.

It has to re-design its platform, correcting the mistakes of the presidential campaign. And it has to overcome the splintering of the left-wing vote that was a major cause of its failure on 21 April.

Talks have been going on this week between Socialist, Green and Communist leaders with a view to fielding a single candidate in key constituencies.

At the local level, however, candidates appear reluctant to fall on their sword, and as the discussions continue, the number of constituencies in question appears to be steadily falling. Last week it was 227, now people are talking of maybe just 100.

Few will be particularly surprised if this pact falls apart before it has even been agreed.