As a gesture to austere times, the cabinet travelled to Chequers on Friday not in individual ministerial limousines, but on a coach. There were other features of the awayday at the prime minister's rural retreat which gave it something of the atmosphere of a school outing. At lunchtime, David Cameron turned tour guide and took around the house those of his colleagues who had never been there before. The prime minister joked: "I'd better do it now, before Nancy and Elwen trash the place." There's usually someone at an outing who doesn't keep to the programme. When they resumed after a coffee break, Ken Clarke was missing. With his colleagues already in session, when the justice secretary finally came through the door he was bellowing into his mobile phone. "Blimey," said the prime minister to laughter all round, including from Mr Clarke. "It took years to get him to use a mobile phone and now we can't get him off it."

The serious purpose of this all-day session was to assess their first 10 weeks in power and prepare for the rockier road ahead. The fuss which erupted when Nick Clegg stood in at prime minister's questions and described the Iraq war as "illegal" – not the Tory position – was a reminder that this is a government welded out of two parties with very different perspectives on a lot of issues. Iraq won't break them apart: that is an argument about history. The real tests of their cohesiveness will be arguments about the future. Several of them are already bubbling to the surface: over university tuition fees, care for the elderly, welfare benefits, and the reorganisation of the NHS. The government is beginning to leak a bit. A row between Liam Fox's Ministry of Defence and the Treasury over the funding of Trident has made headlines. George Osborne and Danny Alexander gave a joint presentation to the Chequers gathering about the spending review. They included a warning to colleagues about not conducting negotiations through the newspapers.

The spending squeeze is the most obvious source of tension within the government. Lib Dems in the coalition report that they find some of the Conservatives far too blasé about just how excruciatingly painful the cuts are going to be. One Lib Dem minister has a mantra for his Tory colleagues: "A £1bn cut is £1,000 taken away from a million people."

Yet it is too crude to see the internal dynamics of this government as simply a case of Tories in one corner, Lib Dems in another. A potentially much more important divide is between coalition Conservatives and coalition Lib Dems on the one side, and non-coalition Conservatives and Lib Dems on the other. This bifurcation is partly about jobs. Government is much more fun for those Tories and Lib Dems sitting in ministerial suites wielding power. That is also a bonding experience for them. Things look rather different to backbenchers of the two parties. They don't have much in common except that they are still on the backbenches. This division between the coalition and non-coalition parts of the government is also about ideology. The unreconciled Tory right distrust their own leader even more than they resent being in government with Lib Dems. The Chequers meeting started with David Cameron giving a joint PowerPoint presentation with Nick Clegg. The prime minister remarked that the coalition required better management of their respective backbenchers. "I know that," he said. "And I know you do too, Nick."

Conservatives ought not to be grumbling about their leader. For them, the coalition is paying dividends. The Tory poll position is substantially improved on their election performance. It is the Lib Dems who have more cause for concern. Their support has slumped. People who voted Conservative have got what they wanted: Tories in government. Many people who supported the Lib Dems were not voting for a Tory-Lib Dem coalition, even though Nick Clegg's statements during the campaign made it clear that this was a likely outcome.

The biggest backlash is in the north of England and Scotland where memories of the Eighties and Nineties still arouse visceral anti-Tory feeling, a loathing now transferred to the Lib Dems for sleeping with the enemy. Nick Clegg has been confronted with this anger in his own Sheffield constituency. Visiting a school recently, the Lib Dem leader was accosted by a man wearing a T-shirt emblazoned with: "Nick Clegg betrayed us."

It is a healthy sign for the coalition that Tory cabinet ministers are sensitive to the potential for this to grow into a big problem for the Lib Dem leader. One Conservative cabinet member who has not got a Lib Dem minister on his departmental team says he's not pleased about this at all. He regards it as "a pain". He is compensating for the lack by reaching out to Lib Dems on the backbenches to be sure that he is alive to their concerns and that they feel involved in his decisions.

Those who see Messrs Cameron and Clegg interacting in private report that the two men are sustaining a good personal chemistry. The prime minister prefers the politics and the company of his deputy to that of many of his Tory colleagues. Many Tory and Lib Dem ministers are building decent relationships with each other. It actually seems to be an advantage that they did not have much, if anything, to do with each other before they went into government together. One reason why New Labour was so riven with poisonous and debilitating personality feuding is that they all knew each other far too well. It was incestuous. Politicians who have been in the same party together for decades come into government loaded with all the grudges, envies, paranoias and other enmities towards colleagues that have been accumulated over many years. Politicians of separate parties working together for the first time are not burdened with all that baggage. One shrewd insider identifies an interesting paradox: two naturally personable leaders have "depersonalised" government. This doesn't mean there aren't arguments. And there are some humdinger rows ahead over spending cuts. It does mean that there may be a better chance that policy arguments can be conducted robustly but without personal nastiness.

Labour's attacks on the government are so far proving counter-productive if the aim is to try to split the coalition. It is actually driving them closer together. After the Cameron-Clegg presentation and the Osborne-Alexander duo, the Chequers meeting was treated to a third double act by Michael Gove for the Tories and Chris Huhne for the Lib Dems. Mr Gove did a little comedy turn about the Labour leadership candidates in which he dubbed David Miliband, Ed Miliband and Ed Balls as "Geeky", "Gawky" and "Grumpy".

The coalition will paint Labour as a party that is irresponsibly unwilling to address the deficit because they won't say how they would bring it down; trapped in the past; obsessed with its own internal wrangles; and in hock to vested interests. It is a game plan encouraged by the behaviour of some of the leadership candidates.

One of the challenges presented by the coalition is how the Conservatives and the Lib Dems maintain their distinctive identities. That is a bigger problem for Nick Clegg. It is partly an issue of visibility. With the exception of Mr Clegg, the Lib Dem members of the cabinet tend to be less visible than the Tories, partly because, with just five seats at the table, there are many fewer of them. There is an interesting question about the party conferences in the autumn. When a party is in government, ministers try to win applause in the hall and media attention for their speeches by unveiling good news. "Conference, I am delighted to announce today that the government will legislate to…"

No one has yet worked out how this should be managed when there are two governing parties with two separate conferences to please. Are the ministers from the Lib Dems, whose conference meets a fortnight before the Tories, to be allocated a party bag of announcement goodies to keep their people applauding? A medium-term problem for the coalition which is already beginning to simmer is what happens when the Tories and Lib Dems become electoral competitors again. That will come to the boil next May. Strategists in both parties are already puzzling about how they will conduct themselves in the spring elections for local councils and the devolved governments in Scotland and Wales. Can they remain friends at Westminster while they are foes on the stump?

For the longer term, there is a conflict of interests. The ultimate strategic objectives of the two parties are not the same. Nick Clegg took the risk of going into coalition with David Cameron for good reasons. The Lib Dem leader hopes to prove that his party is capable of doing government and to acclimatise Britain to coalition politics. The Conservatives are not in this because they want to see Britain perpetually governed by coalition; that prospect would horrify the vast majority of Tories. They did it because it was expedient on this occasion when the voters denied them a majority.

That is the big question which hangs over the coalition in the longer term. It was not addressed at Chequers. Quite understandably. The immediate challenges of the next few months are already formidable enough.