Approaching the offices of Podemos, Spain’s 20-month-old party of the revitalised left, has a touch of the pilgrimage to it. It’s just an ordinary office block in central Madrid, young people beavering about; it could be anyone’s political HQ. But the party itself, the speed of its growth, the radical mould-breaking of its socialist leader, Pablo Iglesias, is significant for left-wing parties everywhere. Like Syriza in Greece, Podemos has already, merely by existing, achieved what was previously thought of as impossible: the capture of popular imagination by some means other than snatching the centre ground. If Pablo Iglesias can win in Sunday’s elections, Jeremy Corbyn can, Bernie Sanders can. More than that, he will have taken the whole conception of “moderation” – as a static political quality that normal people yearn for – and smashed it. That’s why people hang on his every word.

In person, Iglesias is confident and trenchant, but always calm. He looks younger than his 37 years, especially on the roof in the sharp November sun, having his photo taken, yet he has the mature authority of the university lecturer. He speaks as though it comes as no surprise to have people listening intently to him; there’s something faintly Hollywood about the way he describes himself and his trajectory. Talking about the party – which hit a peak of popularity in December of 2014, but tailed off significantly this year – he reaches for a boxing simile. “I always give the example of Muhammad Ali. At the beginning, he boxed, moving quickly and delivering a lot of blows. But in the famous battle in Kinshasa with George Foreman, he learned how to receive blows, learned to withstand many attacks, resisting them to finally win.”

His rise started with La Tuerka, the YouTube political discussion programme he created in 2010. I guess the closest analogy would be with Russell Brand’s The Trews. At that point, he would never have been allowed anywhere near a mainstream Spanish current affairs programme; supporters say he was routinely portrayed as something between an anarchist and a Venezuelan terrorist-sympathiser. But audiences built for the internet channel and, Iglesias recalls, “it was like our gym. In La Tuerka, we trained. In La Tuerka, we allowed our debate to evolve, we learned how to speak to the media, we practised the techniques of communication that allow people to understand you.” It sounds as if he’s watching his political evolution as a montage sequence from Rocky IV.

Iglesias (centre) and the rest of his team during a campaign meeting in Madrid Photograph: Marcos del Mazo/Demotix/Corbis

He describes his political identity romantically, too, tracing its roots back through generations of Spanish activism: “I have participated in political and social collectives since I was 13 or 14,” he says, not preening, nor without pride. “My family is very politicised. My father was in jail as well as my grandfather, who was condemned to death, and finally spent five years in Franco’s jails.”

As a university lecturer, he was politically active in the anti-globalisation movement: at Prague in 2000, Genoa the following year, Gleneagles in 2005. I tell him that I always associated those protests, indeed that time, with political failure, remembering above all the muddle, the lack of a programme, the big, angry “no” to globalisation giving way to a surrendered, “Well, ok then, so long as you promise that corporate capture won’t kill anybody (that I know).” Iglesias disagrees entirely, which is unlike him – his preferred conversational mode is to respond to every question with “exactly” or “absolutely”, a sort of emollient, un-left-ish manner with maybe the faintest whiff of condescension. “In those protests, we understood the importance of globalisation: we understood that a big part of the important decisions weren’t being taken by democratically elected governments, but rather, institutions that weren’t chosen by anyone, like the International Monetary Fund (IMF) or the World Bank. Many of us met through these movements – we didn’t say we were anti-globalisation, we said we were for another globalisation – and have ended up crossing paths again in Podemos or the European parliament.”

Pablo Iglesias: ‘Podemos has changed the political language. Photograph: Juan Carlos Lucas/NurPhoto/Corbis

He is very clear on this: much of what has become plain, and expressed by both left and right, since the financial crash – the dominance of technocrats over democratic institutions, the erosion of the nation state and, with it, meaningful democracy – was already obvious to, and articulated by, that early-noughties protest movement, right down to the inevitability of the crash itself. The fact that it didn’t seem to catch on then is no reflection on its ideological strengths.

As important, in preparation for setting up a party, were his academic influences as a lecturer at the Complutense University of Madrid: Antonio Gramsci, the Italian Marxian, along with Ernesto Laclau and Chantal Mouffe, thinkers whose ideas form the spine of both Podemos and Syriza. Iglesias learned from them that the most important thing is not that activists should have a single, homogenous identity, but that they should be united as to the identity of their enemy. As Iglesias elaborates, “The enemy must have a face. Because when the adversary disappears in a nebulous network of power, or structures of power, it’s much more difficult for people to identify them.”

The frame in Spain is the people against la casta (loosely, “the establishment”) who could be personified in a corrupt politician, a feckless aristocrat or a tax-avoiding corporate CEO. “Podemos,” he says, “has changed the political language. Now everyone speaks of la casta; everyone speaks of change. Televised debates are filled with young people, 20- to 30-year-olds. I think everyone recognises that now nothing will be the same.”

This is not the beginning and end of Podemos’s political vision – Iglesias spells out simply, “We have to protect public services, we have to protect health care, education. We have to avoid leaving unemployed people with no kind of help, we have to create a project for the future. It’s what everybody wants.”

Established in January 2014, Podemos – which translates to “we can” – pulled in 1.2m votes by the European elections in May of that year (nearly 10% of the electorate). By December, it was outpolling the established Spanish Socialist Workers’ Party (PSOE) and was briefly, astonishingly, the most popular party in Spain, overtaking the right-wing Partido Popular (PP) to poll 26%. Iglesias can take much of the credit. He is a wonderful speaker – lines such as “Take your dreams seriously” leap into your memory. But the quote that cropped up all the time at the end of last year – met on the left with a mixture of hope and rue – was this: “When you study the successful experiences of transformative movements, you realise that the key to success is to achieve a connection between the reality you have diagnosed and what the majority actually feels.” It’s typical of the way he writes – his book, Politics in a Time of Crisis, came out in English in November. The language is plain but a little clunky, the mood is optimistic but a little sober, and the point stares you in the face: of course. When you make an observable kind of sense, people start to agree with you. But how, goddammit?

Iglesias with the hashtag #ItIsTheMoment. Photograph: Heino Kalis/Reuters

The party has made and continues to make a success out of owning the 21st century, the landscape of the modern. Iglesias credits this not to his own youth, but to the enduring strength of Spanish leftism. “Change has different expressions,” he explains. “It’s funny – Jeremy [Corbyn] and I have very little in common, whether it’s in age or background, he’s an old leader of Labour, he has lots of parliamentary experience. That’s not the case with me. But still, many of the Spanish media referred to Jeremy Corbyn as the British Pablo Iglesias. How was that possible? Because, in reality, the most important thing isn’t who is bringing the ideas, but the actual ideas.”

To divert momentarily back to British politics, it’s salient to consider the state of flux we’re in: having this conversation in November, I broadly agreed with this, that Corbyn was a different style of politician but that their appeal was similar. Now, I think that Corbyn’s parliamentary and political past is integral both to his appeal and his problems and, as a phenomenon, he’s quite different to Podemos, who are agile and flexible.

Yet, Iglesias says, they share the same contemporary territory: “I think that the ideas that are old are those of Tony Blair. The ideas of Tony Blair have failed and are old ones. And even though Corbyn seems unassuming, and reminds people of Labour’s traditional link to the unions, he brings ideas that can solve problems while the ideas of Blair only made things worse.”

Iglesias is an internationalist and an optimist. He is resolutely upbeat about Alexis Tsipras in Greece: “Many say, ‘But Syriza couldn’t achieve its goals,’ but things started to happen in Europe that can’t be seen separately. Would Corbyn’s victory in the Labour party have been possible if change hadn’t happened in Greece? I think that everything is linked and that parallel movements are being produced. It’s true that adversaries fight and the fundamental project of Germany and many sectors of the European oligarchy was to defeat Tsipras and defeat his government. They weren’t able to.”

The upcoming 20 December general election will see Iglesias and Podemos come up against the Spanish Socialist Workers’ Party (PSOE) leader Pedro Sanchez. Photograph: Jorge Guerrero/AFP/Getty Images

With elections looming five days before Christmas, the party has had to face the fact that its popularity declined in the middle of the year, which Iglesias will allude to, but only loosely – “It’s true that we received many attacks and we made a lot of errors, and this you see. But I’m very optimistic, with respect to December.” Last month saw the numbers start to climb again, but before that, they had dropped below 10%. Some blame Ciudadanos (meaning “citizens”), another new party that shares ground on corruption and workers’ rights, but has a racist top note (no healthcare for migrants) that is proving dispiritingly popular. “I think that a lot of that comes out of nowhere,” Iglesias says rather languidly. “There are many who want to see in Ciudadanos a type of change, but that in reality nothing really changes. I think in that support of big business and media, you really see the intention of the oligarchy.”

Some ascribe the dip to Iglesias himself and his failure to work well with the other parties of the left, though his supporters reject that. (Sirio Canós Donnay, part of the London Podemos circle, explains: “People who say that are talking about United Left – it has really good people, but it’s really old, it has really slow structures. Podemos couldn’t possibly have had a blanket agreement with them, that would not have been democratic.”) Others, still, level the charge that Podemos is fanciful, that their boldest plans – a basic citizens income, debt cancellation – are not realistic.

“The problem with neo-liberalism,” Iglesias begins, unabashed in his use of a term that most politicians still avoid for fear of sounding like an anarchist, “is that it constructs realities that are almost immovable.” He settles in to his explanation and, even though he must have said it often, it sounds passionate and unrehearsed. “In other words, it’s working for its own perpetuation. But, history has shown that everything can change. And I think that we’re moving towards policies that have to do with the democratisation of the economy, against the totalitarianism of the market.” I’m surprised by how serene he looks, faced with the task of overturning the world order. Nobody need raise his voice or wave his hands about, Iglesias’s manner seems to say, for this revolution: it’s inevitable.