ANYONE expecting a cat fight in Bute House on Friday would have been thoroughly disappointed. The meeting between the First Minister, Nicola Sturgeon, and the new Prime Minister, Theresa May, seems to have been more than amicable. Sturgeon even tweeted a picture of them both shaking hands on the steps of her official residence to “show the girls” that nothing is impossible.

But behind the smiles there was very hard politics going on – even a rather male game of poker. May is not prepared to contemplate another independence referendum, which the FM has said is “likely”. But the PM surprised many by announcing that she will not trigger the all-important Article 50 on Brexit until there is a “common UK-wide approach”. This could mean she is using Scotland as a way of keeping Britain in the EU, which would infuriate most of her ministers. (The suggestion that Scotland might have a veto on Brexit aroused fury on social media: “who is running this country, May or Sturgeon” was a common post.)

On the other hand, May's gambit might be an attempt to use Scotland as a means of keeping Britain in the European Single Market (ESM) by forcing her own Brexit ministers to accept free movement as the only way to save the Union. Sturgeon has made clear that remaining in the ESM is her red line as far as the Article 50 negotiations are concerned. Do the Brexiteers care so much about immigration that they would break up Britain over it?

Any way you look at it, this seems a gamble by a Prime Minister who, contrary to her image, seems to like to live dangerously. It is also a risk for Sturgeon. Would her Nationalist supporters thank her for doing a deal on the EU? Or would they prefer Alex Salmond's assertion that there must be an automatic independence referendum after two years of Article 50 being declared?

But independence aside, there did seem to be a curious meeting of minds between the FM and the PM. Perhaps this was because of May's conspicuously social democratic rhetoric during her Downing Street speech last week. A lot of it indeed could have been cobbled from Nicola Sturgeon's election speeches last year with its promises on gender equality and ending the privileges of the wealthy and the privately educated. All very good, but I suspect it will take a lot to persuade Scottish voters that Theresa May is “on your side”.

May may not be Cruella de Vil, but she does bear a striking resemblance to that former Tory matriarch, Margaret Thatcher, not least in the company she keeps. The May cabinet is the most right-wing we have seen in 30 years, with Thatcherite retreads such as Liam Fox, David Davis, Michael Fallon – plus of course Boris Johnson, the thinking person's Donald Trump. Nigel Farage the ex-Ukip leader, was the first to congratulate her on the quality of her ministerial team, which tells you all you need to know.

Others she has drafted into her government may disabuse those who seem to believe that women are naturally more left-wing than men. For example, Liz Truss, the new Justice Secretary, is a co-author of the ultra-Thatcherite Britannia Unchained, which condemned British workers as “the worst idlers in the world”. The Brexit belle, Andrea Leadsom, who believes men should not be child-carers, is a fracking enthusiast who has been put in charge of Environment, despite her evident doubts about climate change. That should be no problem however since May has axed the Department of Energy and Climate Change in her clear-out of what remains of the Cameron green agenda.

Now, there is a theory going round Westminster that the presence of all the Brexit rightists in May’s cabinet is all a clever ruse to "keep her enemies closer". They will have to carry the can if and when the negotiations on exiting the European Union get nasty. That means Fox, Davis et al will have to resign, ha ha. Boris Johnson is similarly being given enough rope to hang himself – though he famously survived doing that on a zip wire during the London Olympics.

I don't buy the idea that Theresa May has a soft centre and is only using the Brexiteers as human shields. She really doesn't like having people around who she doesn't trust or get on with – as she demonstrated by sacking Michael Gove. She has cleared out much of the metropolitan thinking Tories such as Oliver Letwin along with the Notting Hill set of Cameron and Osborne. She represents the return of the narrow tradition of provincial Conservatism, for whom the patron saint is Grantham's Margaret Thatcher.

Making Boris Johnson Foreign Secretary looks like an extraordinary gesture of little England contempt for the rest of the world – and that's precisely how the rest of the world has taken it. When has a British Foreign Secretary ever before been greeted by the international community as "outrageously" behaved and a “a liar with his back to the wall” (as the German and French foreign ministers said of Johnson)? President Obama's state department wasn't impressed either. It remains an astonishing appointment given Johnson's nostalgia for the British Empire, fondness for talking about “picanninies” and for writing offensive poems about foreign leaders. The critical mass of the May cabinet is very much to the right and these politicians have been put in the driving seat. This is likely to provoke confrontations on a series of issues over and above Scotland future relations with the EU.

There is first of all the question of giving security to EU nationals living and working in Scotland. May has refused to promise that they will not be deported, which is damaging not just to Britain's image but Scotland's economic wellbeing. We need these skilled people to grow the economy and the suggestion that they have no right to stay here will discourage many from coming in the first place.

This is part of the wider issue of immigration which Scotland sees very differently to the Brexit bunch in Westminster. Race is not a toxic issue in Scotland as it has been south of the Border where there has been a shocking increase in racial attacks since the referendum. It could become a toxic issue of course, especially if migrants continue to be demonised as job stealers and benefit tourists in the UK media. But the main issue again is economic: Scotland needs migrant workers to fill essential roles in the NHS and to pay taxes to finance public services.

The Scottish Government will also regard with dismay the dismantling of the Climate Change Department. The environment is theoretically devolved, but Scotland remains part of the UK's system of decarbonisation subsidies, which have already been substantially reduced. This could damage the fledgeling non-fossil fuel economy in Scotland.

More immediately, there is Trident, the renewal of which will be debated by MPs in Westminster tomorrow. May has made absolutely clear that the nuclear “deterrent” is sacrosanct and Scotland will continue to house Britain’s weapons of mass destruction – though some of her ministers aren't entirely sure where. The new Tory Leader of the House, Chris Grayling, seems to believe Trident is based in Rosyth rather than Faslane. Tomorrow's Government motion even specifies a “permanent continuous at-sea deterrent”, which removes any possibility of decommissioning or storing the nuclear warheads while keeping the Vanguard submarines operational with conventional weaponry.

Trident is one of the deepest fault lines in UK – Scottish relations after Brexit, and not just because support for renewal appears waning in the opinion polls. Tomorrow's motion will be opposed by 98 per cent of Scotland's MPs, with only the Conservative Scottish Secretary, David Mundell, likely to support it. All the major parties – SNP, Labour, Greens – in the Scottish Parliament oppose Trident renewal. Yet the Defence Secretary, Michael Fallon, has made clear that Trident is a done deal, and the renewal motion will be carried – not least on the strength of English Labour MPs who will defy their leader.

No, the First Minister and the Prime Minister may be happy to have cordial conversations about the value of women in politics. But after the hand-shakes are done they are on a collision course.