(Repeats Sunday story with no changes)
* EU legislature faces challenge to its legitimacy
* Turnout has fallen at each election, but powers have grown
* MEPs want to pick Commission chief, more say on euro zone
* Critics want bigger EU role for national parliaments
By Paul Taylor
PARIS, April 27 (Reuters) - At each election since Europe's unique experiment in cross-border democracy began in 1979, the percentage of voters casting ballots for the directly elected European Parliament has fallen. Yet with each new treaty, the assembly has gathered more powers.
When citizens in the 28 European Union countries vote on May 22-25 for 751 members of the bloc's legislature, opinion polls suggest they will for the first time elect a sizeable phalanx of politicians bent on reversing 60 years of European integration.
Whether the influx of perhaps 150 to 200 Eurosceptics of the hard left and far right will halt the march of the continent's little known and often unloved assembly is far from certain.
While many agree the EU suffers from a "democratic deficit", there is no consensus on whether the solution is to strengthen the parliament or give national legislatures more control.
Incoming EU lawmakers are set to embark from Day One on a new power struggle with the Union's governments.
The first test is selecting the president of the European Commission, the EU's executive body, who will for the first time be chosen under the provisions of the 2009 Lisbon Treaty.
The treaty says the European Council of EU leaders nominate the candidate "taking account of the elections of the European Parliament and after having held the appropriate consultations".
Parliament's main groups think that means national leaders should nominate the leading candidate of the group that wins the most seats - Jean-Claude Juncker if the centre-right is ahead or Martin Schulz if the Socialists are in front.
It is not clear whether the leaders will interpret the treaty in the same way and go along with parliament's wishes or pick their own preferred candidate.
That could prefigure a wider tug-of-war between the EU assembly and national governments and parliaments over the direction of Europe.
"The parliament is the driving force of European integration," says Catherine Trautmann, a Socialist MEP and former mayor of Strasbourg.
"The Commission has everything to gain from drawing its democratic legitimacy from the parliament," she said.
Beyond the struggle over the Commission president, Trautmann expects more battles in the next five years as parliament demands a bigger common EU budget and a say in the euro zone's supervision of national budgets and its bailout fund, created on a strictly intergovernmental basis.
NO EUROPEAN SOVEREIGN
On paper, the parliament is an equal co-legislator with the council of EU ministers on a growing array of policy areas including business and financial services, the environment, health and safety, transport and energy.
It has played a major role in shaping a wave of financial regulation and a European banking union that followed the financial crisis.
Since 2009, it has also had the power to ratify or reject international treaties signed by the EU, a greater say over how the bloc's 135.5 billion euro ($187.5 billion) budget is spent and scrutiny over the cash-guzzling Common Agricultural Policy.
It has flexed those new muscles, notably by rejecting an agreement with the United States on exchanging personal data, which lawmakers decided infringed civil liberties.
Yet it is not quite a real parliament in the eyes of many of its own members or of the EU's main member states.
The fact that voter turnout has dwindled from 62 percent in 1979 to just 43 percent in 2009 is one reason, but not the main one. After all, participation has fallen in national elections, too, and low turnout in local polls does not lead voters to question the legitimacy of their mayors or city councillors.
While some parties tend to send second-ranking politicians to Strasbourg, the assembly also boasts a sprinkling of elder statesmen and political stars such as former Belgian Prime Minister Guy Verhofstadt, leader of the liberal group.
The enduring challenge to its legitimacy has more to do with history and the tenacious survival of European nation states.
The EU legislature is seen as a remote and anonymous body by many citizens whose political universe remains largely national. As former U.S. House of Representatives speaker Tip O'Neill famously said: "All politics is local."
Germany's constitutional court dealt its ambitions a severe blow in a 2009 ruling on the Lisbon Treaty, stipulating that the German national parliament retains ultimate budget sovereignty and should take a greater role in scrutinising EU policies.
The main thrust of the ruling, which denied that the European Parliament was a representative body for a sovereign European people, was that member states confer limited powers on the EU, subject to national parliamentary scrutiny.
"NO TAXATION WITHOUT REPRESENTATION"
Parliaments in Britain, France and the United States all won their spurs by asserting the power to levy taxes.
In 17th-century England and 18th-century France, kings who had exhausted their ability to borrow to finance wars or service their debts had to summon long-dormant assemblies of notables who then grabbed control of their treasuries.
"No taxation without representation" was the battle cry of the American Revolution against British colonial rule.
Yet because the European Union is not a federal state but a community of nation states pooling sovereignty over a limited range of policies, the European Parliament has no tax powers - representation without taxation.
The EU's common budget - equivalent to 1 percent of gross domestic product, while national budgets are on average 40 times that level - is negotiated by member states. The European Parliament has only a take-it-or-leave-it choice at the end.
To some members, this frustration explains the assembly's voracious appetite for regulation and its tendency to support all and any increased European spending.
"This bulimia for regulation comes from a lack of political power," said Corinne Lepage, a former French environment minister who sat in the liberal group in the outgoing parliament and is heading an independent Citizens' Europe list this time.
"We don't have tax-raising powers. We can't amend the budget line by line. We can't initiate legislation," she said.
"Right now, we ought to be in emergency session on Ukraine, but we have no power over foreign policy. If the European Parliament had the power to act and weigh in on that, we wouldn't waste our time on the size of tractor wheels," said Lepage, who spent much of her mandate working on food safety and trying to block genetically modified crops.
The EU budget is mostly financed by national contributions, with only a small part coming directly from customs and excise.
The biggest member states - not just Eurosceptical Britain but core countries Germany and France - oppose giving the EU a direct revenue stream, in the name of national sovereignty. They have also refused to let the European Commission issue debt to fund agreed EU investment projects.
Parliament cannot hold individual members of the Commission to account but only sack the entire executive, which it did once in 1999. Neither the European Council of national leaders nor the Eurogroup of finance ministers of the euro zone, the two most powerful decision-making bodies, is accountable to it.
There is a widespread feeling among lawmakers that they were sidelined from the crisis management procedures and institutions established in haste during the euro zone debt crisis, which they say need to be brought under "democratic accountability".
"The elected representatives of the people cannot go on watching the European Council take decisions as cows watch passing trains," says Sylvie Goulard, a French liberal MEP who campaigns for greater euro zone integration and the issuing of common euro zone bonds.
Goulard and former Italian Prime Minister Mario Monti called in a 2012 book, "Democracy in Europe", for the creation of a euro zone parliament within the European Parliament to exercise scrutiny over decisions limited to the 18-member currency union.
Plans for such a body are already on the drawing board in parliament, French conservative MEP Alain Lamassoure said.
However, creating a euro zone parliamentary forum that had more than consultative status would appear to require a change in the EU's governing treaty.
Despite lawmakers' frustration at the limits on their role, many stakeholders say the assembly has gradually gained in authority and maturity, and they expect that incremental process to continue despite the eurosceptic backlash.
David Earnshaw, a former British Labour MEP who is now chief executive of Burson Marsteller, one of the major public affairs consultancies in Brussels, says the big change in EU politics in the last five years has been the rise of pan-European political parties, in which national leaders form a collective leadership.
He expects that trend to be demonstrated by the election of the Commission president this year.
"Will they betray the electorate? I'm not convinced that Angela Merkel, Francois Hollande, Mariano Rajoy or Mark Rutte will go against the candidates they supported in their political families," said Earnshaw, co-author of a reference book on the European Parliament.
Some EU diplomats are not so sure, arguing that the German chancellor, the French president and their peers will not want to let parliament take the power of nomination away from them.
The arrival of a large intake of eurosceptical MEPs could be an opportunity to bring public debate on the rights and wrongs of European integration inside a chamber that has previously been dominated by a cosy pro-EU consensus.
But the newcomers could also taint parliament's image by using it as a theatre for gesture politics rather than a working legislature.
The insurgent right-wing populist parties mostly advocate leaving the EU and its single currency, closing borders to immigrants and applying trade protectionism.
They may well splinter into several parliamentary groups and have little direct influence on policy, but their impact on the political mood in member states could be greater if they pull mainstream conservative parties towards scepticism.
Charles Grant, director of the London-based Centre for European Reform, sees parliament as a power-hungry, spendthrift cheerleader for a federal Europe, which he argues has too much influence over the Commission, driving unnecessary regulation.
In the pamphlet "How to build a modern EU", published last year, Grant advocated giving national parliaments more say in EU governance, including the power for half of them to veto current or proposed EU laws they deem to infringe national sovereignty.
He also argues that a forum of national parliaments, not the European Parliament, should supervise euro zone governance.
Britain will take a similar line in any negotiation on reform of the EU treaty, but since it is not a member of the euro zone, it is unlikely to prevail. ($1 = 0.7228 Euros) (Writing by Paul Taylor; Editing by Will Waterman)