House of Lords reform is a private coalition grief into which the Labour party must decide how best to intrude. The short-term political gratification for Ed Miliband lies in simply watching the Lib Dem and Tory trains heading towards one another on the same section of track. It will be gruesome when they meet.

Nick Clegg has set his heart on a more democratic upper chamber (by which he means an elected one) as compensation for past liberal sacrifices on the altar of coalition. Conservative backbenchers – perhaps as many as 90 of them – are threatening to rebel against a reform bill in the House of Commons. Then, of course, the peers have to sanction their own demise, which, unsurprisingly, they are not so minded to do.

Further complicating the matter is the question of whether a major change to the legislature should be put to the country in a referendum. Clegg is in no hurry to test public enthusiasm for another Lib Dem-branded constitutional reform, given what happened last time. Besides, the hope is to get Lords reform done fairly promptly. No one in government wants the issue to sprawl all over the parliamentary calendar, sending voters the signal that Westminster takes a more profound interest in its own navel than in real lives elsewhere. The implicit threat of endless delay and filibuster  – torturing Cameron and Clegg for months - is one of the more potent weapons in the Tory rebels’ arsenal.

It is hard to believe that the Prime Minister gets animated by the idea of Lords Reform. Constitutional wonkery isn’t Cameron’s style. If he has to have battles with his own back benchers he’d rather pick them himself than have them forced on him by Lib Dem policy demands. Tory MPs struggle to understand why Cameron feels he needs to help Clegg out, when the junior coalition party plainly has nowhere else to go. Lib Dems, goes this argument, ought to know their electoral place. But that is a misunderstanding of how the coalition dynamic works at quad level. Cameron and Osborne recognise that crucial bits of their programme have been shoved through parliament at considerable cost to the Lib Dems. The balance sheet of who owes what to whom is kept constantly at hand and Clegg’s team feels it has a bunch of favours still to claim after helping Cameron out with tuition fees and NHS reform.

If things get really ugly, the Lib Dems can refuse to pass boundary changes that the Tories feel are needed to iron out a pro-Labour bias in the country’s constituency cartography. That, say Conservatives, would be a low blow.

So Labour can just sit back and watch the coalition tear itself apart, right? Well, not quite. In theory the party is committed to Lords reform too. That doesn’t mean Ed Miliband is duty-bound to side with the Lib Dems. Far from it. The party’s official policy, at least according to its 2010 manifesto, is for a fully elected chamber. There is ample room to trash Clegg’s plans for a partially elected one (with 20 per cent appointed) as another “miserable compromise” – alluding to the Lib Dem leader’s own fatally dismissive characterisation of the Alternative Vote electoral system shortly before he ended up inviting the country to say “Yes” to it in a referendum.

Labour are already making supportive noises around the idea of a referendum on Lords reform. There are solid constitutional reasons for wanting to do this, but it is hard to escape the feeling that it is all really mischief to cause difficulty for the government and prise open coalition divisions. Fair enough, really. That is partly what oppositions are meant to do.

Nonetheless, it hasn’t escaped the attention of some Labour MPs that a more sophisticated way to achieve the same goal would be to vote with Lib Dem MPs in the Commons. That would have the added benefit of demonstrating a commitment to the principle of a more democratic parliament. Even if reform clears the Commons it will still run into huge difficulty in the Lords, but then the Lib Dems’ pain will be more conspicuously the responsibility of the Tories. In other words, Labour could use the process as a test bed for a slightly more collaborative relationship with the third party. With the odds currently favouring a hung parliament at the next election, Miliband might want to practice a slightly more accommodating stance towards the Cleggites.

Lib Dems are certainly watching Labour’s behaviour around Lords reform with that in mind. No one is seriously suggesting there is footsie between the two parties, but voting on the same side would mark an important symbolic threshold crossed. A Lib Dem cabinet minister told me recently that “Labour could make or break Lords reform” adding, for good measure, that even if it proves impossible to get an elected chamber on the statute books “why not let the Tories be the ones that f-- it up?”

Why not, indeed? One reason is that many Labour MPs are sincerely opposed to the kind of Lords reform on offer. They fear for the primacy of the Commons and resent the creation of  a bunch of swaggering senators with 15-year terms, who will claim a democratic mandate without facing routine re-election battles.

Another reason is that Labour MPs still find it hard to skip an opportunity to spite the Lib Dems. Hating the Tories is work, goes the joke, while hating the Lib Dems is pleasure. And the temptation to engineer a government defeat in the Commons – always a supremely newsworthy event – is enormously hard to resist. “We’re not in the business of throwing life lines to Clegg,” is what one Labour strategist says. “They can’t see past us,” is how one ally of Lib Dem leader bitterly responds.

And in the background are all the recriminations over the failed “Yes to AV” campaign. Miliband’s side say Clegg’s toxic brand lost it. Lib Dems say Miliband was too weak to whip his party in favour of a measure that the leader endorsed.

It is certainly true that no one now sees the national vote against AV as a triumph for anyone but the Tories. Some Labour MPs who fought against the change did so on principle; plenty did so out of hatred for Clegg; a couple just wanted to destabilise Ed Miliband. In any case, the net result was that the Labour party looked weak, divided, hostile to reform and incapable of rising above tribalism, while George Osborne celebrated a campaign victory. It was not the opposition’s finest strategic hour. They should examine that record carefully when thinking about how to play their cards over Lords reform.