There are three certainties about British life: it rains on bank holidays; England's chances at the Euros are always inflated and then disappointed; and the right claims that the BBC suffers from "leftwing bias". When it was announced on Friday that the TUC's senior economist Duncan Weldon would become Newsnight's new economics correspondent, it was like catnip for conservatives. Weldon's first crime is to work for the TUC. Trade unions may be Britain's biggest democratic movement, but they are generally shunned or demonised by the media and political elites alike. His second crime is – like many journalists – to have a political background; in his case, he once advised Harriet Harman. Anonymous senior Tories are briefing that it is a "grade-A BBC stitch-up" and that Arthur Scargill would have been "a more objective appointment", while the Conservative MP Andrew Bridgen has issued a letter of complaint to the corporation.

The Daily Mail is at the forefront of the campaign to prove the BBC is a den of socialism, and has even attacked Sherlock Holmes as "more evidence" of the corporation's "leftwing bias". The Mail on Sunday's Peter Hitchens claims that the BBC "is hopelessly morally, socially and culturally biased against conservative ideas". Ironically, I recently discussed the issue with him when he interviewed me for a BBC Radio 4 documentary he was presenting. Earlier this year, George Osborne's private secretary wrote to the BBC director general to complain, among other things, that I had been introduced on the BBC news channel as a "social commentator" rather than a "high-profile leftwing activist." It was amusing, not least because I am routinely introduced as a "leftwing firebrand", but have yet to hear the likes of Hitchens or the Times's stridently Conservative Tim Montgomerie described by BBC presenters as "rightwing firebrands".

It is a campaign based on myths and deception, but it is extraordinarily clever. It allows the right to police the BBC: to make the corporation fearful of crossing certain lines, and to ensure that the right sets the political agenda. Leftwingers are reluctant to return fire for fear they will help to fatally undermine the BBC. After all, its existence is refutation of the dogma of "private good, public bad", and much of the right would like to privatise it. The Murdoch empire, only temporarily cowed, is always circling: a few years ago, James Murdoch attacked the "dominant BBC" and called the scope of its activities "chilling". But the left's reticence is symptomatic of a wider phenomenon – of a right with few scruples about going on the offensive, while the left adopts a relentlessly defensive posture.

The truth is the BBC is stacked full of rightwingers. The chairman of the BBC Trust is Chris Patten, a former Conservative cabinet minister. The BBC's political editor, Nick Robinson, was once chairman of the Young Conservatives. His former senior political producer, Thea Rogers, became George Osborne's special advisor in 2012. Andrew Neil, the presenter of the BBC's flagship political programmes Daily Politics and This Week, is chairman of the conservative Spectator magazine. His editor is Robbie Gibb, former chief of staff to the Tory Francis Maude. After the BBC's economics editor Stephanie Flanders left for a £400,000-a-year job at that notorious leftwing hotbed, JP Morgan, she was replaced by its business editor Robert Peston. His position was taken by Kamal Ahmed from the rightwing Sunday Telegraph, a journalist damned by the Guardian's Nick Davies for spinning government propaganda in the run-up to the Iraq war.

BBC stalwart John Humphrys last week joined the chorus of voices alleging "liberal bias" at the BBC. Here is a man who was slapped down by the BBC's own trust last year for violating impartiality and accuracy guidelines in BBC2's The Future State of Welfare. It was an extraordinarily biased piece of TV that fuelled widespread myths about social security. With such coverage, this "liberal-biased" BBC shares the blame for leaving the public completely ill-informed, with, for example, voters estimating that 34 times more money is lost through benefit fraud than is actually the case.

Tory politicians favour the BBC as a useful recruitment service too. After Andy Coulson was driven from No 10, David Cameron replaced him with the then BBC news controller Craig Oliver. Boris Johnson's former communications supremo was the former BBC political correspondent Guto Harri; after moving to News International in 2012, he was replaced by the BBC's Westminster news editor, Will Walden.

Rather than having a leftwing bias, research actually suggests the BBC's output is biased towards establishment and rightwing sources. A study by Cardiff University academics found that while there is always a bias towards political incumbents, the ratio in favour of Conservative politicians appearing on BBC news is far greater than it was in favour of Labour figures when Gordon Brown was prime minister. Business representatives appear much more than they do on commercial news, and appear 19 times more frequently than trade union voices on the BBC Six O'Clock News.

When the financial system went into meltdown, BBC interviews were dominated by City voices like stockbrokers and hedge fund managers, rather than critics of a sector that had plunged the country into disaster.

Take the privatisation of the NHS: it is barely given any coverage by the BBC, and when it is, it is repeatedly presented on the government's terms. When the legislation was voted through, the BBC reported it as "Bill which gives power to GPs passes."

The same goes for foreign issues, too. When Israel invaded Gaza in 2009 the BBC provoked uproar when it refused to give the Disasters Emergency Committee appeal any airtime. It was left to the late Tony Benn to read out the number on air, while BBC journalists who asked to put their names to the appeal were privately warned they would be sacked.

In part, this is the legacy of the BBC's bruising battle with New Labour over the Iraq war, which led to its chairman, director general and journalist Andrew Gilligan being driven from the corporation. The episode left the BBC supine and fearful. Its news output is deeply reactive, rather than agenda-setting, structured along the lines of government announcements.

For too long, the right has got away with weaving a fairytale of BBC leftwing bias. Until the left starts complaining – and loudly too – the BBC's agenda will be shaped by supporters of government, big business, the free market and western foreign policy. That does not just subvert honest journalism: it undermines our democracy.