Much has been written about the sociological dimensions of the referendum. It split the country down the middle, pitting older, working-class voters in post-industrial towns and cities, deprived seaside resorts and the agri-business rural areas of England and Wales, against the young, middle-class and-university-educated cosmopolitan centres. Older cleavages re-emerged, too, with the fractured politics of unionism and nationalism visible again in Northern Ireland and Scotland.

It is as if the country, a week ahead of the centenary of the Battle of the Somme, unconsciously recalled an age when politics turned on the axes of class strife and the future of the nation, and arguments raged about the rise of German power and Britain’s place in the world.

These ley lines of socioeconomic division will matter for Brexit as much as they did for the referendum itself. Britain now faces a prolonged period of uncertainty in which the politics of economic and social class interests will come to the fore in a struggle over the terms of Brexit. At stake is who will win and lose in the new settlement established between Britain and Europe, and within the United Kingdom itself.

The dominant sectors of the economy – in particular the City and financial services – will seek above all to protect their integration in the single market. Political leaders who are desperate to shore up a declining tax base and prevent operations shifting into the eurozone will come under immense and unrelenting pressure to defend the City, further cementing the UK’s dependence on financial services. Likewise, multinationals operating across a number of European sites will fight hard for the preservation of single market status. They cannot afford the delay, obstruction and cost of immigration and customs controls.

Islands of prosperity that benefit significantly from EU funding, such as the university towns, will advocate strongly for their interests, as will the farmers dependent on EU funds, and businesses in the low-wage sectors that rely heavily on the free movement of labour. Each will have trade bodies and sector organisations to mobilise on their behalf. Political geography will matter, too. Scotland has a powerful, well-led government to press its case for staying in the EU, and independence for Scotland may yet become an argument for the status quo. Wales can look to its government to defend the funding that Europe has poured into its former mining valleys. London’s mayor will become a tribune of the City.

Who then will speak for the working-class Brexit heartlands of England? There is no English Parliament to give them voice. Patchwork devolution to the cities and counties of England has only just begun to spread economic and political power across the country, and mainly to the big conurbations that voted to stay in. Trade unions are barely visible in the private sector of low-wage England.

As we become more reliant on the “kindness of strangers” to pay our way in the world, the question of whether businesses invest in post-industrial England will become more pressing. And as economic growth slows and Britain’s fiscal position deteriorates, the heaviest burden may yet fall on low-income families, as economists warned before the referendum.

The governor of the Bank of England Mark Carney at a press conference last Thursday. Photograph: Matt Dunham/Reuters

The Conservative high command has sensibly junked the government’s surplus rule for the public finances, but the pressure on government spending will be intense. Under the guise of drawing investment back to the UK, the drumbeat for cutting corporation tax, deregulating the labour market and shrinking the welfare state will soon start in earnest. Without countervailing pressure, the Brexit working class will pay for their votes in cuts to public services and tax credits.

Labour is the historic political arm of these workers, but it is a severely depleted political force, facing extinction as a credible party of government. If it can emerge intact from the Shakespearean carnage in which it is engulfed, it will confront a double bind. If it advocates staying in the single market as a member of the European Economic Area, which is critical to workers in manufacturing companies such as Nissan and Airbus, it will have to concede free movement – unless the EU is prepared to contemplate immigration reforms it did not offer to David Cameron. But this is the last thing most Labour MPs want to do. The party’s metropolitan strongholds are proud of their diversity, and the welcome they give to migrants, but its post-industrial heartlands are not. They voted for an end to free movement. Ukip will offer it if Labour does not.

The referendum debate showed once again that immigration is viscerally divisive; it would be wishful thinking to pretend that the political cleavage it represents will diminish any time soon. This is in part because the recent migration surge will be carried into the population mix of the 2020s, but also because demographic and fiscal pressures reinforce the case for continued inflows. The referendum result doesn’t change that underlying truth.

The Labour party’s metropolitan strongholds are proud of their diversity, but its post-industrial heartlands are not

There is clearly widespread support for increased public spending on schools, housing and GPs in areas experiencing high migration flows. But more is needed if Labour is to play its part in shaping the Brexit settlement. Unless it is to be trapped in a zero-sum game with Ukip on free movement, it will have to revisit the political economy it practised in the pre-crisis era.

Croslandite tax-and-transfer will always remain a part of the social democratic toolkit, but it cannot be a substitute for increased public capital investment and regional economic strategies that mobilise resources to boost infrastructure, skills and innovation in post-industrial towns as well as the burgeoning cities, with substantial devolution of political power to support this agenda.

The fractured Labour coalition of middle-class liberals and working-class social conservatives can at least find points of unity on economic policy, while the more protracted task of creating new forms of English identity, in which migrants, the cosmopolitan young and the so-called “left behind” can each find a home, might then start.

The Conservative leadership contest looks set to crystallise the Brexit choices that the referendum did not flush out: whether we stay in the single market, and hope that the EU will offer free movement reforms, or seek a free trade deal or WTO status, and an end to free movement. Absent a general election or a second referendum, only 150,000 members of the Conservative party will get to make that choice.

So Labour, as well as progressive allies in other parties, should make a democratic, as well as economic case, for the country’s future, insisting either that these choices are put to the people in an early general election, or that Parliament should decide the terms on which Article 50 is triggered, reserving the right to “withdraw the withdrawal” and hold a second referendum on whatever Brexit deal is struck with the EU.

The terms of Brexit will be shot through with competing political and social class interests, just as the referendum was: they should be contested openly and democratically.

Nick Pearce is professor of public policy at the University of Bath