Jane Boyle went to the match with the boyfriend she was due to marry five days later. She died when the crowd stampeded in terror and she fell underfoot. John Scott, who was just 14, was so badly mutilated it was at first thought that he had been bayoneted to death. Thomas Ryan was kneeling down, whispering a prayer into the ears of another dying man when he was himself shot. Two little boys, one aged 10, the other 11, were among the dead.

The 14 people who were killed at Croke Park stadium in Dublin on 21 November 1920 were far from the only victims of the Troubles of 1916 to 1923 that led to the foundation of the Irish state. Indeed, 31 people in all were killed on that single day alone. Yet those killed when troops and police opened fire on the crowd at the headquarters of the Gaelic Athletic Association (GAA) are remembered more clearly than many of the others. They died in the most traumatic of a concentrated series of violent incidents. There was another Bloody Sunday during the more recent Troubles in Northern Ireland, but this was the day for which the term was coined.

In the early morning, 14 secret agents, the core of the British Intelligence operation against the Irish Republican Army, were killed in their suburban Dublin homes by a squad organised by the IRA leader Michael Collins. In the evening, three prisoners, two of them senior IRA men, were killed by the British "while trying to escape".

But it is what happened in the afternoon that makes this month's visit by the Queen to Croke Park, the headquarters of the GAA in Dublin, so resonant. Tipperary and Dublin were playing a game of Gaelic football in front of a crowd of 5,000 people. Croke Park was surrounded by a mixed force of armed members of the Royal Irish Constabulary, regular troops and members of the Auxiliaries, an irregular force largely recruited in England and attached to the Irish police to help fight the IRA. Armoured cars blocked the exits from the grounds. The intention was that all the spectators leaving Croke Park would be searched for arms.

Military and police participants later claimed they were fired on by someone in the crowd. Whether or not this was true (and there was no independent inquiry), what happened next is broadly clear. Over the course of a few minutes, the police and Auxiliaries fired 228 shots, and an army machine gun at one of the exits fired 50 rounds. Fourteen civilians were killed, two of them trampled to death in the panic. Sixty more were injured. The secret military inquiry, which became public only in the past decade, concluded that the firing was "carried out without orders, and was indiscriminate and unjustifiable". The almost universal view among Irish nationalists was that the killings were a deliberate reprisal against unarmed civilians for the assassinations of the intelligence officers earlier in the day.

One of the victims of Bloody Sunday was the Tipperary player Michael Hogan. The Queen will meet GAA members under the Hogan Stand of the monumental new Croke Park stadium, rebuilt in the 1990s for a capacity of 82,000. That amateur sports played in just one country can fill such a stadium is extraordinary. Much more extraordinary, though, is that the stadium can now play host to a British monarch.

The rebuilt 82,000-seat Croke Park. Photograph: Ray McManus/Sportsfile

Even a decade ago, the idea would have been unthinkable. Now, the only official comment on it from the GAA is a discreet notice on the Croke Park website, concerning arrangements for the museum at the stadium, where the Queen will spend 45 minutes: "The GAA Museum will be closed from Saturday 14 to Wednesday 18 May inclusive." The studied pretence that nothing much is happening is itself testament to the reality that the unthinkable is coming to pass.

A century ago, if you asked a typical Irish nationalist what was distinctively Irish, they'd have listed the big forces that defined their culture: the Catholic church, nationalist politics, attachment to the land, the Irish language and the GAA. Today, almost all of those markers of identity are gone or weakened. The church may never recover from the child-abuse scandals that have destroyed its authority in the past decade. The Fianna Fáil party that captured mainstream nationalism and dominated Irish politics for half a century was decimated in February's election. Ireland has long since ceased to be a rural, agricultural society. The Irish language clings on but the aim of making it the everyday tongue is further from fulfilment than ever.

The one part of the package that still functions is the GAA, which is not merely surviving but thriving. If you want to give a foreign visitor a quick sense of something unique to Ireland, you bring them to Croke Park for a game of Gaelic football or, better still, hurling.

In spite of the glamour of professional sports such as soccer and rugby, the GAA's showpiece inter-county championships, played out over the five summer months, account for 60% of all attendances at sporting fixtures in Ireland. The vast majority of those fans also follow British football teams, such as Manchester United, Liverpool or Celtic, and many are passionate about, for example, the Munster rugby team. But the GAA touches a very different nerve. In a world where global sporting spectacles are packaged for passive consumption, the GAA appeals to something local, intimate and democratic. It doesn't just belong to Irish people, it gives them a sense of belonging.

The laureate of the GAA, Tom Humphries, captured this perfectly when he wrote: "The GAA player who performs in front of 70,000 at the weekend will be teaching your kids on Monday, or he'll be selling you meat or fixing your drains or representing you in court. The soccer player who performs in front of 70,000 people at the weekend will be moaning about too many games and trying to sell you his personalised brand of leisure wear."

The GAA evokes feelings that go so deep you can be completely unaware of them until something happens to reveal their power. The most recent revelation came in early April, when dissident republicans murdered a young policeman, Ronan Kerr, in County Tyrone. Kerr was a Catholic and a member of his local GAA club, the Beragh Red Knights. In killing him, the dissidents violated a community's sense of itself, the pride it takes in the young men and women who play on its local GAA teams.

The hero of the classic GAA novel, Charles Kickham's Knocknagow, published in 1873, is a farm labourer who goes on to the hurling field with the cry: "For the credit of the little village!" GAA players still take the field for the credit of all the little villages – not just the literal ones like Ronan Kerr's Beragh, but the psychological villages to which we cling in a globalised culture – the idea of a place, of a community, of something that is not yet owned by a TV company or a corporation.

Ronan Kerr's coffin is carried through his home town. Photograph: Paul Faith/Press Association

Ronan Kerr's funeral produced an image that is in its own way even more powerful than any that will be captured at the Queen's visit to Croke Park: the pictures of his GAA team mates and the Tyrone county manager, Mickey Harte, passing his coffin from their shoulders on to those of his police colleagues. It was a picture of the dissidents' worst nightmares. The GAA was defining the police in Northern Ireland as "us" and Ronan Kerr's killers as "them". There is no other institution in Ireland, north or south, that has the authority to do this.

Kerr's funeral and the Queen's visit both point to the GAA's ability to grasp something that can be very difficult for organisations rooted in notions of tradition. Ideas of place, of community, of identity are hugely important, but they are not static. What has been remarkable about the GAA in recent years has been its capacity not just to respond to change but to create it.

The Queen's visit to Croke Park may have been planned only in recent months, but it is the culmination of a process that has been under way for more than a decade within the GAA. Very calmly and quietly, a series of the GAA's elected presidents, such as Joe McDonagh, Sean Kelly and Nickey Brennan, have set about modernising the organisation. That meant, in political terms, aligning it more closely to the mainstream of Irish nationalism, which had been disgusted by the IRA's violence and which hankered for ideas of Irish identity that were positive and open rather than embittered and embattled.

Change was driven from the Republic, but the leadership was careful not to alienate the more conservative membership in Northern Ireland. The GAA's democratic structures were a big help – the conservatives had their say and were never allowed to claim they had been railroaded. Bit by bit, the GAA took down the barriers that protected the old exclusive attitudes. It lifted the ban on its members playing other sports. It opened up Croke Park to the "foreign games" of rugby and soccer whose infiltration of 19th-century Ireland it was founded to oppose. It allowed God Save the Queen to be played on its hallowed turf in 2007 when the England rugby team came to play Ireland. (It helped that the money the GAA made from renting out Croke Park was channelled back to its own local clubs.)

Most importantly, in 2001, the GAA deleted its Rule 21, which barred members of the Northern Ireland police or armed forces from joining. It is worth recalling that, at the time, the delegates to the GAA congress from five of the six Northern Ireland counties voted to keep the rule in place. Within a decade, those same people were embracing Ronan Kerr as one of their own. The GAA managed a shift in attitudes so gentle that it took a terrible murder for everyone to realise that a quiet revolution had taken place.

The shift was hard for two reasons. One is that the GAA was embattled for a long time, especially in Northern Ireland. The grounds of the famous Crossmaglen club in south Armagh, regular winners of the All-Ireland club football championship, were occupied by the British Army from 1974 until 1999. There were arson attacks on GAA clubhouses by loyalist paramilitaries. GAA members were easily identified as Catholics and therefore made clear targets for sectarian killers. In 1997, for example, Sean Brown, chairman of the Wolfe Tone club in Bellaghy, County Derry, was kidnapped as he locked up the clubhouse after a match and murdered shortly afterwards.

But the GAA has also been psychologically embattled. Its games have a very long history, but also a long history of disparagement.

In 2009, when the Cork county goalkeeper Donal Og Cusack came out as gay, the novelist Colm Tóibín hailed him as "the first gay hurler since Cúchulainn", a tongue-in-cheek reference to the Iron Age mythological warrior-hero whose feats with stick and ball make him the precursor of today's hurlers. The odd thing about this mythology is that it doesn't seem at all ridiculous. If you watch a top-class game of hurling, the speed, strength, dexterity and personal courage on display do remind you of warriors in the hurly-burly of war before armour and technology.

But hurling has not always been appreciated. Arthur Young, the English economist, who witnessed a game in the 1770s, called it "the cricket of savages". A 1936 MGM documentary movie called Hurling was advertised with the slogan "Shillelaghs in Swing Time as 30 wild Irishmen demonstrate their game of athletic assault and battery". In John Ford's 1957 movie The Rising of the Moon, an English tourist, seeing injured players carried by on stretchers, asks nervously: "Charles, is it another of their rebellions?"

The question was not entirely stupid, for the GAA undoubtedly was part of a cultural rebellion that could not be cleanly separated from a military one. In its own mythology, the GAA saw itself, in the words of IRA leader Harry Boland in 1919, as having "drawn the line between the garrison and the Gael" – separated the native Irish from English influence. The GAA was effectively taken over by the revolutionary Irish Republican Brotherhood (forerunner of the IRA) in the late 1880s. More moderate nationalists subsequently regained control, but the IRB influence remained very strong. It was not entirely illogical for the police and army to attack Croke Park on Bloody Sunday – senior figures in the GAA at the time included IRA leaders like Boland, Austin Stack, Eoin O'Duffy and Michael Collins himself.

It's not surprising, therefore, that it's taken a long time to completely disentangle the GAA from violent republicanism. But there is another reason why the shift has been slow: it is quietly audacious. For the task that the GAA will complete when the Queen visits Croke Park is a momentous one – that of creating a distinctive and proud Irish identity that is not anti-British. The GAA came out of a time when the easiest answer to the question "what does it mean to be Irish?" was "not British". It takes real courage to replace that easy negative with something more positive and fluid.

But there is something pleasingly neat in the way it is happening. The irony is that the GAA is a quintessentially Victorian institution. It is a classic creation of the late 19th-century English drive to codify sports with written rules and centralised organisations. The men who established the GAA in 1884 saw themselves as traditionalists and cultural nationalists, preserving the ancient games of the Gael from the new vigour of rugby, soccer and cricket. But their reaction took the form of emulation – they did for Gaelic football and hurling what the English were doing for other sports.

There is a further irony that the Queen might appreciate, however. Not only is the GAA a classic Victorian organisation, it has been much more faithful to its origins in late-19th-century sporting culture than the English sports that influenced it. If you want to get some sense of the ethos of English sport before the rise of professionalism, without the snobbery that went with it, the best place to look is probably the GAA. The Queen will find many of the notions that characterised the old Corinthian spirit – character, community, playing for the sake of it – alive and well and living in Croke Park.