The prime minister and his new chancellor may have come to a harmonious agreement over their special advisers but there remains a significant point of friction between them: whose alma mater is superior?

The rivalry between Old Wykehamists and Old Etonians played out in the Commons last year when Jacob Rees-Mogg (Eton) jibed at Nick Boles (Winchester) during a Brexit debate. “My honourable friend makes a characteristically Wykehamist point: highly intelligent, but fundamentally wrong,” Rees-Mogg said – to the bafflement of most voters, though perhaps not most MPs.

But the feud has now moved to the top of government with the appointment of Rishi Sunak as chancellor. He attended Winchester College in the 1990s; his next-door neighbour Boris Johnson enrolled at Eton in 1977.

According to Sunak, his parents – a GP and a pharmacist – “sacrificed a great deal so I could attend good schools … That experience changed my life and as a result I am passionate about ensuring everybody has access to a great education.”

The chancellor is one of four former Winchester College boys in the House of Commons: the others are the Tory MPs Alex Chalk, Marcus Fysh and John Whittingdale. Boles resigned from the Conservative party over Brexit and did not stand in the 2019 election. There are 11 Old Etonians on the Tory benches, according to the Sutton Trust, out of 150 (41%) who went to independent schools.

In contrast, only 14% of Labour MPs were privately educated. But Old Wykehamists can count two of Jeremy Corbyn’s closest aides – Seumas Milne, his strategy and communications chief, and James Schneider, his former director of communications – among their number.

Alumni of Winchester College are known as Old Wykehamists after William of Wykeham, bishop of Winchester, who founded the school in 1382. The college has produced a large number of archbishops and bishops, along with a strong showing for politicians (including the Labour leader Hugh Gaitskell and Tory-turned-fascist Oswald Mosley), writers, actors and cricketers. One notorious Old Wykehamist is Rurik Jutting, who is serving life for raping, torturing and killing two women in Hong Kong.

Nevertheless, the school claims to prepare boys “for a future of compassionate leadership”.

“Winchester does not produce a ‘type’,” its website says. “Instead, we instil a respect for the individual and a keenness to contribute to the community.” The motto is “Manners makyth man”; annual fees are £41,709.

One former Winchester boy told the Observer that Wykehamists had “an inferiority complex, but they also think they’re cleverer and politer and more thoughtful and less entitled”.

“They think that this explains why they might end up as civil servants – or backroom operators like Milne or Schneider – rather than on the frontbenches. So I think a lot of them might be taken aback, almost affronted, by Rishi Sunak’s rise and obvious ambition – it’s a real kick in the teeth for that self-image as the sort of people who could run the country if they wanted to, but simply chose a different path.”

He recalled an “environment where paranoid homophobia and racism and gross jokes about working-class people – cleaners referred to as ‘skivs’, for example – were commonplace. And all part of a culture where taking offence or calling it out as repellent was seen as a sign that you weren’t sophisticated enough to grasp the levels of irony at work.”

Sunak, he said, would “probably have met girls through a house dance about once a year (to which it is likely he would have smuggled miniature hotel bar bottles of whisky in his underpants) and in school plays.”

In 2017, Winchester College was forced to defend its role in an alleged cover-up of serious physical abuse at Christian summer camps attended by its pupils in the 1970s and 1980s. It emerged that the school had known in 1982 about claims of sadomasochistic abuse at the hands of John Smyth, a British QC who ran a series of Christian summer camps attended by its pupils, but did not report the claims to the police.