As only the second woman to hold Britain’s highest political office, comparisons with the first seem inevitable. We know that Ken Clarke certainly sees something of a parallel between the two leaders. “Theresa is a bloody difficult woman, but you (Sir Malcolm Rifkind) and I worked for Margaret Thatcher!” Mrs May herself appears to have embraced the caricature. Addressing Conservative MPs ahead of last Thursday’s leadership ballot, she seized Clarke’s candid description to her advantage: “Ken Clarke may have found me to be a bloody difficult woman. The next person to find that out will be Jean-Claude Juncker.”

It won’t be long until Mrs May puts her negotiating skills to the test. Whether you voted to remain in the EU or to leave it, the reality is that we need a steely negotiator to get the best deal possible for the country as a whole. In this regard, a touch of the Iron Lady about her will be to all of our benefit. However, as the instinctively cautious Mrs May is no doubt herself aware, the new prime minister needs to tread with extraordinary care. It would be an understatement to say that Margaret Thatcher was a divisive figure. Britain doesn’t need a divider right now, if ever. It needs a unifier.

When Margaret Thatcher stood outside 10 Downing Street on 4 May 1979, she quoted St Francis of Assisi before proceeding through the famous black door.

Where there is discord, may we bring harmony.
Where there is error, may we bring truth.
Where there is doubt, may we bring faith.
And where there is despair, may we bring hope.

We’re still divided on her legacy, whether she changed Britain for the better, or whether she left it for the worse. Still, even her staunchest admirers and her fiercest critics must surely agree on one point: whatever she delivered, it wasn’t harmony.

We are at a turning point in our country’s history, and none of us know what’s around the corner. There is doubt and, among much of the 48% who voted to Remain in the referendum, there is a lingering sense of despair. Among other things, we need faith and we need hope. Our future relationship with the European Union will continue to dominate the prime minister’s agenda over the coming months – and years. Invoking Article 50 won’t suddenly make Brexiteers out of Remainers, just as it won’t give Brexiteers an appetite for compromise with our EU neighbours.

Arguments about Europe will rumble on in new forms. Against this backdrop, however, the referendum has highlighted other profound divisions within our country: between young and old, between those who have done well from globalization and those who haven’t done nearly as well, between England and Wales and Scotland and Northern Ireland. People have different ideas about the UK’s place in the world. We remain an unequal country where the economy simply doesn’t deliver for everyone. And the very Union behind our United Kingdom is in doubt.

These raise deep questions about who we are as a country and our place in the world: Should we be more open, or should we take a step back? How do we build an economy where we really are all in this together? How can Scottish, Northern Irish and interests be balanced against Welsh and English interests? In short, how do we keep our country together?

These are questions that will be hovering over our new prime minister from Day One, today. There are signs that Mrs May understands that she inherits a country divided along multiple lines. Now buried beneath the dramatic news of Andrea Leadsom’s departure from the leadership race, on Monday Theresa May issued a veiled critique of George Osborne’s economic record: “We must make Britain a country that works not for a privileged few but for every single one of us.” These are, of course, mere words, but by uttering them at all Mrs May could be signaling a different approach in the government’s approach to reducing inequality. Let’s watch this space.

Mrs May, to date, has said less about how she plans to bring the different parts of the UK together: how to stave off an independence referendum in Scotland and how to maintain Northern Ireland’s open border with the Republic of Ireland. Like her negotiations with the European Union, and like dealing with an unequal Britain, the unity of the UK is not just a background issue but will be riding on everything her government does. That will require a genuine effort to listen and engage with the UK’s devolved administrations to an extent not previously seen on any other issue.

In Scotland, Nicola Sturgeon will naturally be a key player, but perhaps more important will be the role of the energetic and charismatic Ruth Davidson, leader of the Scottish Conservatives. Sturgeon has only an ambivalent interest in making the negotiations work for Scotland; she doesn’t think Westminster can or will understand Scotland. Davidson, by contrast, has a much more genuine interest; she needs Westminster to understand it. As the most senior Tory north of the border, she has a lot to lose if Scotland ends up marginalized. Mrs May should therefore listen very carefully to her perspective. In Northern Ireland, it is much less clear who can advocate its sensitive interests clearly and robustly. The First and deputy First Ministers are unlikely to provide a unified voice, at least for the foreseeable future, while the Secretary of State Theresa Villiers appears unjustifiably blasé about the potential impact of Brexit on Northern Ireland. Perhaps it will take a new Secretary of State for Northern Ireland’s leaders and citizens to feel adequately represented around the Cabinet table and beyond.

Uniting the country will not be the task of one woman, but she will have a critical role to play in that task. To put it another way, many things will be outside Mrs May’s control, but at the very least she must use the influence she does have to avoid unnecessarily disuniting the country. Mrs Thatcher’s vow to bring harmony where there is discord carries more than a touch of irony. Harmony was not one of her legacies. Whatever parallels exist between the two women, Mrs May would do well not to make that one of them. We need harmony, and I hope she will make that her top priority when she is sworn into office as Britain’s 54th prime minister. I wish her very well.