There are angles from which our own faces are unfamiliar. Even in the age of the selfie, it is hard to picture the person that others see when they look at us. We cannot observe ourselves unawares; we will never see ourselves in the mirror with our eyes closed.

This phenomenon applies on a national scale. The country we know ourselves to be is not always the same as the one that other countries observe. The remote perspective can be instructive, even if only to illuminate cultural disparities. On holiday in the US earlier this summer I met a friendly off-duty cop who was very agitated by my government’s refusal to allow law-abiding citizens to carry deadly weapons. Not even knives! But that’s Europe for you; funny Old World.

Crossing the Atlantic is an efficient treatment for the delusion that Britain is significant in some way distinctively from its status as a European country. Viewed from America, the English Channel is not a formidable maritime border. The many historical, linguistic and cultural threads that seem to connect Britain to the US are mere decorative tassels around the edge of the vast continental canvas.

Our preoccupation with their politics isn’t even perfunctorily reciprocated. American TV allots as much coverage to Labour’s leadership contest as the BBC might give to a declining opposition party in a small faraway land. In other words, none. Some newspaper commentators peer at Jeremy Corbyn’s battle with Owen Smith as if it were a nasty rash on the left cheek of British democracy, symptomatic of some political vitamin deficiency; not serious, but worth monitoring.

Our continental neighbours are more emotionally invested in UK politics. Earlier this week, Sweden’s main liberal daily, Dagens Nyheter, ran an editorial casting Labour as the most tragic case in a Europe-wide crisis of social democracy, distinguished by middle-class activists prioritising radical abstractions over strategies for regaining power. The likelihood of Corbyn winning a general election was hardly worth a mention. “Venezuela’s socialist adventures have always meant more to him than the British working class,” the paper wrote.

“Some [US] newspaper commentators peer at Jeremy Corbyn’s battle with Owen Smith as if it were a nasty rash: not serious, but worth monitoring.” Photograph: Danny Lawson/PA

Foreign reporting can irk the natives whose affairs are being too casually summarised, but the succinct dispatch from overseas is often pithily insightful. When Andrea Leadsom pulled out of the Conservative leadership race, Le Monde’s news report noted how the latest British developments “confirm the unbelievable level of unreadiness and irresponsibility of the Brexit supporters”. Viewed from France, the salient point about Theresa May becoming prime minister was how it expressed failure on the part of prominent leave campaigners to assume moral ownership of the referendum result they had engineered.

Conservative Brexiteers made a Faustian pact with anti-immigrant populism. They mostly deny it, of course

The job of extricating Britain from the EU plunges May deep into the gap between Britain as it sees itself and Britain as the world sees it. For many Tory Brexit campaigners, the ideological impetus was never isolationist or chauvinist. The theory was that Britain should become a hub of global trade, woven seamlessly into Europe’s single market while unbound by its meddlesome regulations. But that was a fantasy, unavailable in reality and hard to pitch to a mass audience. So to secure the result they wanted, Conservative Brexiteers made a Faustian pact with anti-immigrant populism. They mostly deny it, of course, because it was shameful.

Many leave supporters insist they were only standing up for democracy; no offence intended. But our European neighbours interpret the result as a spasm of unfriendly nationalist reaction, because that is what they saw from not very far away.

May now has to come up with a deal that assuages the anger that was cynically stirred in pursuit of a Brexit vote, while the stirrers-in-chief want no responsibility for the nasty stuff. Liam Fox and Boris Johnson bicker over who should be in charge of jetset trade diplomacy, happy to sit in business class while someone else does the job of locking migrant nurses out of the NHS.

Meanwhile, the prime minister has to persuade EU leaders that her country is not turning its back on them; that it offers something in return for favourable exit terms. Yet nothing but the sound of doors slamming will persuade a large portion of the British electorate that true Brexit has been achieved. There is no solution to this conundrum in the usual menu of second-tier trade partnerships – variants of the Norwegian or Swiss models. The best economic argument that the remain campaign failed to communicate to voters was that no EU satellite arrangement combining market access and influence over the rules could rival the one we already have as full members.

May’s negotiations have to go well beyond the list of current benefits that Britain might salvage. She needs a broader agenda to discuss what we have to offer as a strategic partner to the European project. The UK’s assets are denominated in power, hard and soft, from our capital city’s role in global finance to our status as a nuclear-armed Nato member.

But the pitch in Brussels, Berlin, Paris and Warsaw must aspire to more than tariff elimination. It must include promises to engage with continental challenges: Isis-inspired terrorism, Russian territorial expansion, energy security. The question is not whether we can cobble together a thin facsimile of EU membership but how the UK can be an upstanding friend and neighbour to the EU after we leave. That is the terrain on which the best deal is struck.

Brussels-baiting Euroscepticism and the internal dynamics of the Tory party have for a generation set the terms of the debate about Britain’s status as a European nation. Yet history, geography and global diplomacy dictate that we can be no other kind. A stark irony of the Brexit vote is that it demands of the prime minister a more thorough, ambitious and diligently applied vision of Britain’s strategic relationship with the rest of the continent than has been possessed by any of her predecessors since Ted Heath. Leaving the EU will involve diplomacy as ardently and explicitly pro-European as that required for joining in the first place. In the intervening years we have spent a lot of energy arguing about what Europe means to Britain. It is time to consider what the rest of Europe might see in us.