By Steve Scherer and Naomi O'Leary

ROME (Reuters) - Italian Prime Minister Enrico Letta has ordered an investigation into the deportation in May of the wife and daughter of fugitive Kazakh oligarch and dissident Mukhtar Ablyazov as questions about its legality multiply.

The mysterious operation in which Ablyazov's wife Alma Shalabayeva and their six-year-old daughter were seized and whisked off to Kazakhstan on a non-commercial flight two days later has put pressure on Interior Minister Angelino Alfano.

Letta told parliament on Wednesday that differing versions of exactly what happened after police raided Ablyazov's Rome villa on May 29 "made further clarification unavoidable".

He said the results of the investigation would be made public, adding that "doubts and shadows will not be tolerated".

This month, a court in Rome annulled decrees prosecutors had issued to authorize the raid and said it was "perplexed" at the speed of the deportation, which contrasted markedly with the normally sluggish pace of the Italian bureaucracy.

Ablyazov, 50, was not at home when police raided the villa, which his family had occupied since September 2012, but he has accused Kazakh President Nursultan Nazarbayev of pressing Italian authorities to "kidnap" his family.

Lawyers for the family say Ablyazov's wife and daughter had valid residency permits issued by Britain and Latvia and that both were forcibly deported despite requesting political asylum.

A former government minister in Kazakhstan, Ablyazov fled the oil-rich Central Asian state after his bank BTA was nationalized and declared insolvent in 2009.

He was granted political asylum in Britain in 2011 and says his life is in danger.

BTA has brought fraud charges against Ablyazov and his allies. Accused of embezzling $6 billion, he has been in hiding since last year when he fled Britain after missing a contempt of court hearing at which he was due to be jailed for 22 months.

The Kazakh foreign ministry has said Italian authorities detained Shalabayeva because she presented a false passport and she was then deported at the Italian government's request.

As well as creating implications outside Italy, the affair has brought trouble for Letta's fragile left-right coalition government, with Alfano, secretary of Silvio Berlusconi's center-right People of Freedom party, in the firing line.

Opposition lawmakers and members of Letta's center-left Democratic Party have pressed Alfano to explain the circumstances of the arrest and deportation for which he was ultimately responsible as interior minister.

Italy's state-owned oil giant Eni is one of several co-owners of Kazakhstan's Kashagan field, the world's biggest oilfield discovery in more than 40 years.

Italian media have also reported that Berlusconi, the head of Alfano's party, is on good terms with Nazarbayev, who has ruled his vast nation of 17 million for more than two decades.

Nazarbayev has overseen market reforms and foreign investment inflows that have ensured rapid economic growth, but he has tolerated no dissent.

Ablyazov, a theoretical physics graduate who earned a fortune after the Soviet Union's demise, told Reuters in an interview in December that he would run for office if free elections were called when Nazarbayev's rule ends.

(Editing by Alistair Lyon)