For many Greeks, Orthodox Easter is a chance to see friends and family, to eat good food or to worship. But for the neo-Nazis in Golden Dawn, who only recently made the switch from "Hellenic" paganism to a professed love for Christianity, it has been an opportunity for propaganda. Last Thursday, the party made headlines with its attempt to stage a "Greeks-only" food distribution in Athens's Syntagma square. The next day, when Athenians were driving back to home towns and villages, Golden Dawn members held open motorway toll booths – which have become a symbolic point of resistance against the rising cost of living in the wake of austerity – so cars could pass for free.

Such stunts have become common for a party that seeks to exploit anger at Greece's social crisis, along with the undercurrent of racism that has accompanied it. As one Golden Dawn voter in the Athens suburb of Petropolis put it to me when I visited Greece last month, she saw them as the only party who would make politicians take responsibility for their "lying and cheating" against the people. She's not alone: almost a year after the elections that saw Golden Dawn shoot from obscurity to the third-largest party in Greece's parliament, it maintains a steady 10-12% in the opinion polls.

What is new, however, is that in the case of Syntagma, the event was prevented from going ahead in the square. A truck carrying food intended for distribution was blocked from entering the square by riot police, who then used teargas – a treatment usually reserved for leftwing demonstrators – to disperse about 50 Golden Dawn members who had assembled for the handout. Later, a Golden Dawn MP, Giorgos Germenis, tried to punch the Athens mayor, Giorgos Kaminis, who had requested the police intervention, as he moved to shut down a Golden Dawn stall selling Orthodox Easter candles to children. Germenis also reportedly reached for a gun – something he has since denied doing.

If the incident revealed something of the true face of Golden Dawn – cheap populist tricks, backed up by violent threats – it stands in sharp contrast to the general pattern. Its members frequently stage "Greeks-only" food handouts and blood donation drives, and they are rarely challenged by the authorities, despite the blatant racism of such initiatives, in a country where violence against ethnic minorities is on the rise.

This fits with a wider atmosphere of impunity. Klio Papapantoleon, a lawyer who has represented victims of assaults committed by Golden Dawn members, told me that the Greek justice system had been "unusually lenient" in judging them, while clients and witnesses had often been "obstructed and encumbered by police officers" when trying to pursue a complaint.

Encouraged by this treatment, Golden Dawn has been doing its best to sink roots into Greece's institutions, building networks of support inside the police force, entering hospitals and, perhaps most worryingly, trying to win teenage recruits by spreading its ideas in schools.

A recent Council of Europe report concluded that Golden Dawn's well-documented role in perpetrating racist violence meant it could be banned under existing laws, yet for now the Greek government seems unable or unwilling to act, preferring to mimic its rhetoric: in February, 85 MPs from New Democracy, the largest party in the coalition, signed a motion that called for anybody not of the "Greek race" to be barred from joining the country's police and armed forces. This contrasts with a recent crackdown on leftwing groups, which has included raids on squats, the closing of Athens Indymedia and the ongoing terrorising of villagers in the north of Greece opposed to a gold mining project.

Yet while journalists understandably want to draw attention to the threat Golden Dawn poses, every piece of sensationalist media coverage reinforces the party's deliberately crafted image. The violence it inspires is real enough, but Golden Dawn is far from being in a position of power. Its activist base remains small; it can not mobilise supporters in large numbers; and its rallies often take place unannounced, so that anti-fascist activists do not have time to gather and chase its members off the streets. The food handouts, staged mainly for the benefit of the media, pale in comparison with the network of solidarity initiatives like the "potato movement" – markets that allow farmers to sell their produce directly to customers, at around 30% less than supermarket prices – or volunteer-run medical clinics, or free after-school tuition for children, that are helping Greek people cope with the impact of mass unemployment and falling salaries. By contrast, as a member of Solidarity4All, a national network that co-ordinates such initiatives, described it to me, Golden Dawn's handouts are a grim affair: "They buy the food, they make everyone listen to 30 minutes of political speeches, then they make everyone wait in line. There's no co-operation."

What's more, many Greeks are simply disgusted by the presence of fascists on their political scene. They are becoming increasingly vocal about this, in public displays of solidarity with immigrants, as they did in a anti-fascist protest in Athens on 19 January, backed up by demonstrations outside Greek embassies around the world. Elsewhere, the expression has been more blunt: last month in Chania in Crete, angry residents threw the party's parliamentary candidate into the sea. International pressure has even forced the Greek government into making noises about tackling the problem, but it is at grassroots level where Golden Dawn is being opposed most effectively, and where it will ultimately be defeated.

One Golden Dawn member I interviewed last year, on condition of anonymity, put the party's appeal to me succinctly: "We do what others don't dare." This is posturing, and it can be broken. But it has to be broken now, before it's too late.