He may or may not welcome it, but Tony Blair has had lavish praise heaped on him by the uber-conservative of US politics, Dick Cheney.

In his autobiography published on Tuesday, the self-declared Darth Vader of the Bush administration pays tribute to the former Labour leader. Not only was Blair America's greatest ally during the Bush years, says Cheney, but his speeches about the "war on terror" were some of the most eloquent he had been privileged to hear.

George Bush's friendship and closeness to Blair have been well documented, but the position of his vice-president, who earned a reputation for secretiveness while at the White House, has been less clear until now.

In the 565-page In My Time, Cheney is unrepentant about the most controversial decisions taken by the White House, from the waterboarding of Guantánamo Bay detainees to the invasion of Iraq.

Recalling a trip to Europe in March 2002, a year before the invasion of Iraq, Cheney says: "I began my trip with a stop in London to visit one of America's closest and best allies in the war on terror, British prime minister Tony Blair. I have tremendous respect for Prime Minister Blair," Cheney writes. "He is a Labour party liberal and I am a conservative Republican, and we didn't always agree on strategy or tactics. But America had no greater ally during our time in office. His speeches about the war were some of the most eloquent I've been privileged to hear."

Meeting at Downing Street, Cheney, an early advocate of invading Iraq, said a decision had not yet been made, but invasion was on his mind: "The president wanted to be absolutely clear that if he decided to go to war, we would finish the job. We would remove Saddam Hussein, eliminate the threat he posed and establish a representative government."

The vice-president even made phone calls to lobby Tory MPs on Blair's behalf. On the eve of the crucial Commons vote in 2003 that authorised the war in Iraq, he writes: "At the request of the British, I had called a number of the Tories, including Iain Duncan Smith, the Conservative leader. He was, on this issue, a rock of support for Blair."

In his memoir, A Journey, Blair praised Cheney, although he said Cheney was wrong to ignore the role of ideas, rather than simply military might, in defending the US. Blair wrote: "He believed, in essence, that the US was genuinely at war; that the war was one with terrorists and rogue states that supported them ... of course, this attitude terrified and repelled people. But ... I do not think it was as fantastical as conventional wisdom opined."Cheney writes that on some things he differed from the former Labour leader. Blair wanted resolution of the Israeli-Palestinian conflict to be a priority; Cheney was sceptical: "I was not as confident as Blair that solving this crisis would take the steam out of the terrorist threat."

Later that year, Blair, Bush and the then US secretary of state, Condoleezza Rice, discussed a plan to get UN weapons inspectors back into Iraq. Cheney, known for his bluntness, writes: "I didn't buy it. It seemed fanciful to me."

Nor did he support Blair when in January 2003 the PM pushed for a second UN resolution. Cheney thought it was a mistake, and says "going to the UN again would make us look hesitant and uncertain". The resolution never materialised.

He regards the invasion as justified, seeing Iraq as a nexus between terrorism and weapons of mass destruction. "With the benefit of hindsight – even taking into account that some of the intelligence we received was wrong – that assessment still holds true," he says.

In the book, Cheney devotes many of the photographs to shots of himself with his grandchildren and tells anecdotes about his dog, Dave. But that does little to soften his image as an uncompromising conservative. In his final chapter, he sums up with a declaration of patriotism. "I am a firm believer in America and its work in the world … We have stood firm in the face of evil and defied history in the selfless way we have done it. Instead of seeking empire, we have sought freedom for others."