"We don't do God," Alastair Campbell famously insisted when journalists pressed the former prime minister, Tony Blair, on matters of faith.

But it appears that New Labour's high command missed a trick by declining to talk up their religious convictions, for new research suggests they would have been preaching to the converted: people with faith are far more likely to take left-of-centre positions on a range of issues, including immigration and equality.

The research, revealed in a new report by the thinktank Demos, undermines the widely held view that members of religious groups are more likely to have conservative tendencies.

Its findings are exciting the interest of senior Labour politicians, who believe that attracting the faith vote will be crucial to the party's fortunes.

The Faithful Citizens report also has implications for the aspiration of prime minister David Cameron for a "big society". It finds that people who identify with a faith are more likely to volunteer, be politically engaged and to become active citizens in their neighbourhoods.

Prominent left-of-centre figures who have strong religious beliefs, like the comedian Frank Skinner, who has spoken out in favour of higher taxes and is a committed Roman Catholic, are rare.

But the Demos report suggests that the example of the outgoing archbishop of Canterbury, who combines deeply held progressive beliefs with his religious convictions, is not unusual.

"Rowan Williams may be far more representative of the religious community than many have suggested," said Jonathan Birdwell, the author of the report. "Progressives should sit up and take note. Their natural allies may look more like the archbishop of Canterbury than Richard Dawkins."

The report found that 55% of people with faith placed themselves on the left of politics, compared with 40% who placed themselves on the right. The report also suggests that people with faith are more likely to value equality over freedom than their non-religious counterparts. It discloses that 41% of people with religious views prioritise equality over freedom, compared with 36% of those without faith.

The report, based on an analysis of the European Values Study, also finds evidence that people who belong to a religious organisation are more likely to say they are very interested in politics, to have signed a petition and to have participated in a demonstration.

Writer and psychologist Oliver James triggered a debate by suggesting that religious people were less likely to be leftwing. His comments came as Cameron attempted to reinforce his party's religious credentials by suggesting that there was "something of a fightback going on" among Christians. "The values of the Bible, the values of Christianity, are the values that we need," Cameron said.

Labour is now also keen to promote its faith credentials. "The progressive cause is often cast as being in opposition to the religious one," Labour MP Stephen Timms notes in a foreword to the Demos report. "Faith group members will be key in any future, election-winning, progressive coalition... Labour can draw new energy and inspiration from engaging with faith groups."

But politicians may find that cultivating the faith vote will become less important in the future. Religiosity among younger citizens appears to be declining, with nearly two-thirds of 18- to 24-year-olds claiming that they do not follow a religion, compared with under one third aged 65 and over.

The writer and philosopher Alain de Botton – whose latest book, Religion for Atheists, examines the consolations of faith for those who do not believe – argues that the internal dynamics of religions often confer progressive views on their followers, who find themselves at odds with today's free-market society.

"The progressive side of religion springs from their frequent reminders to live for others and to concentrate more on the wellbeing of the group than on the happiness of the individual," de Botton said. "In this sense, religions run counter to the implicit philosophy of modern consumer capitalism."