* PM Cameron conducts biggest jobs reshuffle since 2010
* Eurosceptic Philip Hammond becomes Foreign Secretary
* Right-wing ally Michael Gove sacked as Education Secretary
* William Hague resigns as UK's top diplomat
* Cameron pitches Lord Hill as European Commissioner
* Cameron promotes women to address gender imbalance issue (Updates with Hammond, Fallon, other appointments)
By Andrew Osborn and Guy Faulconbridge
LONDON, July 15 (Reuters) - British Prime Minister David Cameron on Tuesday pushed through his biggest government shake-up since coming to power in 2010, promoting women and Eurosceptics to senior roles ahead of a national election in May next year.
In one surprise development, William Hague, Britain's most senior diplomat for the past four years, voluntarily stood down allowing Cameron to appoint Philip Hammond, the defence minister and a prominent Eurosceptic, to the influential post.
In another, Michael Gove, a longstanding Cameron ally and one of his party's most prominent right-wing ideologues, was sacked as education secretary.
Women will now make up six of the new 23-person Cabinet, compared with three of 22 before.
Hammond's appointment immediately stoked speculation that Cameron, the leader of the ruling Conservative party, was trying to give his part of the coalition government a more Eurosceptic flavour to please a vociferous wing of his own party and to counter an electoral threat from the anti-EU UK Independence Party which won European elections in Britain in May.
The choice of Hammond sends a powerful signal to Britain's European allies. In 2013, he said that if the European Union failed to change and failed to agree new terms for Britain's membership then he would rather leave the bloc.
Meanwhile, the shake-up, or reshuffle as it is traditionally known, saw the Ken Clarke, a minister without portfolio and the Conservative's government's most pro-EU member, sacked.
Cameron has promised to try to reshape Britain's EU ties if re-elected next year before giving voters a membership referendum, something opinion polls show could be close.
Cameron appointed Jonathan Hill, a lord, to become Britain's next European commissioner. Hill, who is not well-known, had previously coordinated the government's business in the upper house of parliament.
Hill has voted against deeper European integration in the past, but Nick Clegg, the deputy prime minister and leader of the pro-EU junior coalition Liberal Democrats, had to agree his appointment.
Clegg's aides had previously said he would not approve anyone with overly strong Eurosceptic views.
Michael Fallon, formerly a junior energy minister and someone who has said his party should campaign for Britain to leave the EU if it doesn't get major reforms, was appointed as the country's new defence secretary.
One of the biggest surprises for Britons however was Cameron's decision to sack Gove, whose radical reforms to the education sector have provoked anger among teachers.
He will instead become the Conservative party's chief whip, a role that entails ensuring lawmakers vote in the way the party leadership wants.
Hague, the outgoing foreign minister, will assume a more junior role that will involve him coordinating the government's business in the lower house of parliament. He said he would also step down as a member of parliament in May next year.
"There is a balance between experience on the one hand and renewal. Parties and governments need renewal and we are fortunate in our party to have some extremely talented people now coming to the fore so let's give them their opportunity," Hague told reporters.
Lagging the opposition Labour party in the opinion polls by between three and seven percentage points, Cameron promoted a raft of women to senior posts to correct a perceived gender imbalance.
Nicky Morgan, minister for Women and Equalities, was appointed education secretary, while Liz Truss was appointed as secretary of state for environment, food and rural Affairs in the place of Owen Paterson.
The Labour party has repeatedly criticised Cameron for what it says is his "women problem" - a relative lack of females in top cabinet jobs.
Labour on Tuesday called Cameron's jobs reshuffle the "massacre of the moderates," saying the changes reflected his weakness and failure to reform his party.
"In a desperate attempt to shore up his support within his own Party, he undertakes a reshuffle which has seen centrists kicked out while right-wingers have been promoted," it said.
"This massacre of the moderates shows the extent of David Cameron's weakness." (Additional reporting by Kylie MacLellan and William James; Editing by Jeremy Gaunt)