Whatever people think of Jeremy Corbyn’s supporters, it’s probably fair to say there is an assumption that they are young. That has been the frame since he was elected leader of the Labour party a year ago – the idea that millennials previously turned off by politics were now thronging to the party.

Explaining the rise of old white male socialists in politics – Corbyn in Britain, Bernie Sanders in the US – the millennial focus makes some sense. Both politicians speak squarely to this young demographic, with policies such as the abolition of university tuition fees, universal healthcare or affordable housing (as Corbyn raised in today’s prime minister’s questions). Both understand that this generation is hobbled by neoliberal policies.

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But last week’s YouGov Labour leadership poll shows that, within the party at least, this isn’t the whole picture. Corbyn has a 61% approval rating among 18- to 24-year-olds, 64% among 25- to 39-year-olds and 63% among 40- to 59-year-olds. Meanwhile, of those who have now joined the Labour campaign group Momentum, only a third of those disclosing their age are millennials. And the Guardian’s own research, speaking with 100 Labour officials across Britain in January, showed that those signing up to the party in their thousands were in fact a mix of young and old, first-time members and returnees.

So it seems white-haired democratic socialists also attract supporters from Generation X, those aged 35-50: me, that is. Actor Mark Ruffalo also falls into that demographic. Earlier this year, he interviewed “Bernie from Brooklyn”, exploring how the politician’s upbringing in this New York borough shaped his policies. One part of this video interview jumps out: “You know what you make me feel?” the 48-year-old Ruffalo asks Sanders. “That all of my ideals as a young person can actually grow up with me.”

This, in so many ways, encapsulates the appeal of older leftwing politicians to the over-35s. But we, just like the millennials, are dismissed as naive and unrealistic, and derided by political pundits who insist socialism is dead and that only swing-voter-friendly, centrist politics can win elections, even while the centre-left as a political force has collapsed across western Europe, from Spain to Greece to the Netherlands.

Overusing words such as sensible and pragmatic, the commentariat insists that only a sort of middle England-friendly, marginal seat-focused compromise can walk into No 10. But Generation X felt the ravaging effects of such centrist compromises during New Labour’s 13 years in office, when stealth-privatisation, via PFI, in the NHS and schools, the embrace of a deregulated free market, the failure to support the manufacturing sector and the perpetuation of Tory-style welfare reforms all sucked the life out of public services, while draining the wealth out of the economy.

For my generation, the politics proposed by Corbyn are a long-awaited challenge to the assumption of third-way pragmatism as the natural order of things (even as this natural order leads to financial catastrophe and rampant inequality). Corbyn’s approach also punctures the dominance of marketing and spin – over policies and substance – that seem to go hand in hand with these supposedly “pragmatic” politics.

Meanwhile, older socialist politicians are living rebuttals to the idea that we “grow out of” leftwing politics. In reality, there is no truth to this: researchers have been unable to show conclusively that we become more conservative as we age. One workable theory is that older, more conservative cohorts polled on political views today came of age during a period of political conservatism: they are simply holding on to a position, not shifting it.

If the patronising complaint hurled at millennials is that they are too young to remember the bad days of the old left, my generation does remember. First, we remember that “left” is not synonymous with “bad” and that left politics do have popular support (just check the figures in support of renationalisation of utilities, for example, or fairer taxation). Meanwhile, we also recall Corbyn speaking out against apartheid at a time when the current party of government was calling Nelson Mandela a terrorist and doing business with an internationally boycotted South Africa. Like Sanders, Corbyn was opposed to the Iraq war at the time, foreseeing its devastating consequences for the Iraqi people. And it is striking how often this war comes up when speaking with Corbyn supporters. This isn’t purely about remembering the invasion. It’s about wanting politicians to take a different, more credible approach across the Middle East. In this context, the significance of Corbyn’s apology for the war on publication of the Chilcot report was vastly overlooked by the British media.

It is easy to accuse Corbyn of being stuck in the past, but such throwaway putdowns miss the merits of having consistently held values that “grow up with you”. At a time when trust in Westminster is so eroded, a politician who has repeatedly, Google-verifiably kept to the same principles is going to appeal across generations.