UPDATE 1-Cameron, Clegg relaunch UK coalition after losses
* Cameron, Clegg defend coalition
* Conservative rebels angry with PM's compromises
* Hacking inquiry likely to cap difficult week for Cameron
BASILDON, England, May 8 (Reuters) - British Prime Minister David Cameron rallied behind his coalition with the Liberal Democrats on Tuesday, hoping to revive its fortunes after big losses in local elections, but tempting the wrath of an unruly Conservative right.
Cameron and Lib Dem Deputy Prime Minister Nick Clegg renewed their coalition vows in a rare joint appearance at a tractor factory, pledging to see through an unpopular austerity plan and to get a shrinking economy back on track.
Voters dished out a thrashing to both the Conservative and Lib Dem parties in nationwide local council elections last week, venting their anger after Britain fell back into recession after two years of uncompromising spending cuts.
The loss of support rallied rebels on the right of Cameron's party who blame his failure to win an outright parliamentary majority on his centrist, modernising approach and who fear his compact with Clegg will alienate right-wing voters at the 2015 election.
Speaking alongside Clegg, Cameron said the coalition was committed to "a tough task getting even tougher", clinging to a plan of cutting Britain's large budget deficit.
"We're finding it more difficult to get our economy recovering," Cameron said. "It is tough right now for families to make ends meet.
"We formed a coalition two years ago to try and deal with these problems and I believe the need for that coalition - two parties working together to solve the problems we have in our country - is as important and necessary today as it was two years ago."
Clegg, whose party has plummeted in popularity since forming the coalition, said his party was also prepared for a further two years of spending cuts to balance the books.
"Dealing with the deficit is a means to an end," he told workers at the factory in the town of Basildon, 25 miles (40km) east of London. "We suffered a socking great big heart attack at the very centre of our British economy ... It's painstaking work recovering from that."
Much has changed since a sunny May afternoon in 2010, when a smiling Cameron and Clegg shared the stage in the garden of the prime minister's office in Downing Street for a show of unity after joining forces to end 13 years of Labour Party rule.
Cameron's austerity plan, opposed by Labour and some leading economists, has been extended to seven years from an initial five, the economy is shrinking and households are struggling to square high inflation with meagre wage rises.
CALLS FOR CAMERON'S HEAD
Conservative lawmaker Nadine Dorries warned Cameron at the weekend he could be deposed by anti-European Union party members angry with his courting of the pro-European Lib Dems.
A win for charismatic Conservative Boris Johnson - a man who has his eyes on leading the party one day - in London's mayoral election took the sting out of last week's election thumping. But it also reminded the party it has a successful politician in its midst who shares some of its more traditional views.
Cameron, confident he still has the support of his party's mainstream, ruled out any change in policy on Monday, saying the lesson of the poll defeats was "not about tacking right or moving left".
In a coalition that has pledged to stay together until 2015 in the national interest to deal with Britain's debts, the Lib Dems are also struggling with their political consciences.
Support for the party has been in freefall since 2010, with some of its grassroots members despairing at Clegg's decision to join a Conservative-led government.
There has been speculation the party could break away from the coalition before 2015 to give itself a fighting chance of winning back some of its more left-wing voters.
Cutting an increasingly glum figure, Clegg said the parties "came together to rescue, repair and reform our economy", but admitted "it is only right that we now take stock".
With both men unwilling to use taxpayers' money to stimulate a struggling economy, the government will hope the Bank of England will on Thursday commit to more measures to boost demand by buying assets with newly-created money.
However, economists do not expect the BoE to act, having slashed interest rates to a record low and pumped 325 billion pounds into the financial system.
Cameron can expect more flak this week when two former allies appear at a judicial inquiry into a phone-hacking scandal at one of Rupert Murdoch's British newspapers.
Rebekah Brooks and Andy Coulson, who both served as News of the World editors, will appear at the Leveson inquiry on Thursday and Friday.
Coulson moved from the paper to become Cameron's spokesman, while Brooks was a friend of the Conservative leader.
The hacking scandal has prompted searching questions over the close relationship between police, politicians and the press, with Cameron vulnerable to accusations of poor judgment in his dealings with the two former Murdoch executives.
British media have reported Brooks was willing to release text messages and emails between herself and the prime minister.