The Labour Party makes a lot of people think about the Cold War, not least because Jeremy Corbyn looks as if he believes the wrong side won that one.

His horrible reluctance to accept that an attack on a Russian dissident using a Russian nerve agent might just have something to do with Russia has deepened and demonstrated the huge gap in the world view of the Labour leader and many of his colleagues. Hence the latest talk that some Labour MPs are again thinking about quitting the party to set up a new outfit.

Tensions between Corbyn and Labour colleagues who remember the Tony Blair years with pride not shame have been simmering since he became leader in 2015 but like the Cold War between the West and the USSR, it never quite boils over into all-out war.

But while it’s tempting to compare the internal politics of the Corbyn Labour Party to the Cold War, it’s not accurate. The Cold War had only two main players. Within Corbyn’s party, there are three powers, all engaged in political combat against each other.

The battering of the Blairite centrists by pro-Corbyn forces is the most eyecatching sub-plot and could yet blow the whole thing up if some of the MPs disgusted by Corbyn’s record on Brexit, anti-Semitism and now on Russia finally decide enough is enough.

But those pro-Corbyn forces are in fact members of two distinct groups who both despise the centrists but don’t agree about much else.

These factions are Momentum, the headline-grabbing “grassroots” socialist movement that brought many new members into Labour for Corbyn, and the trade union movement that created the party.

The Momentum hipsters and the union brothers have already fought one battle, over who gets to pick the next Labour general secretary (the official who runs the party bureaucracy), with Momentum backing down before the fight went nuclear.

But if that row was about internal office politics, the two sides are about to square off for a fight that will actually mean something to people in the real world, because it’s about sex, or gender, or both.

How should Labour, and the law, treat transgender people? Current laws allow someone born male to become legally female, but only with the approval of doctors. Some people want to change those laws so someone can basically decide their own gender just by filling in a form and regardless of whether they have made any attempt to change their body or lifestyle. Advocates, many of them young Momentum “bros” who idolise the Guardian’s Owen Jones, say that’s the only way to properly respect the identity of transgender people.

Sceptics, many of them long-standing Labour feminists, worry that allowing men unlimited freedom to take on the legal status of women could erode the legal protections and safe spaces set aside for those born female; domestic violence refuges, women’s prisons, sport and statistics could all be altered for the worse, they fear. You might have seen reports of women “self-identifying” as men and requesting access to male changing rooms at a swimming pool in Dulwich to highlight the issue.

The immediate political flashpoint is Labour’s all-women shortlists. Some in the party want places on those lists to be open to “self-defined women”, a term which appears to include transwomen who are still, in one interpretation of the current law, technically male. Feminists have cried foul at this, mounting a legal challenge to see if the party’s proposed position on all-women shortlists would be lawful.

Several of those feminists are trade unionists, women who grew up with socialism and feminism as twin creeds. It’s often missed, but the changing nature of the workforce and the union movement means that the typical trade unionist today is not a brawny factory hand but a middle-aged woman working in the public sector.

One of the feminist groups raising questions about self-defined gender is A Woman’s Place UK, which organises public meetings to discuss the issue — or tries to: its activists, like other women who question gender law reform, say they face persistent abuse and harassment from some trans-rights campaigners.

The women behind Woman’s Place are trade unionists, a fact that has caused significant tensions on the Labour Left. Supposedly progressive supporters of self-identified gender and “self-defined” women are brutal in their criticism of women they dismiss as “trans-exclusionary radical feminists”. Some Labour women have been expelled from the party; others have quit in protest.

But Women’s Place feminists have a powerful ally. Unnoticed outside Labour circles, they have been supported — in the Morning Star, no less — by Andrew Murray. He is chief of staff to the Unite union, an adviser to Corbyn, and, with him, a founder of the Stop the War movement whose anti-Western world view is visible in Corbyn’s Russian equivocation. In short, he’s a big deal in Corbynworld.

And Murray has backed those feminist trade unionists who have dared to challenge the Momentum-driven orthodoxy on gender laws, insisting they have a right to be heard and warning their opponents: “This attempt to silence women’s voices does a disservice to us all.”

The first skirmishes of Labour’s gender war have already resulted in one political casualty: Munroe Bergdorf, a transgender model, quit a Labour advisory panel over offensive comments about lesbians and feminists, shortly after her position was questioned by John McDonnell, the bone-crunching shadow chancellor. Meanwhile, the BBC reported yesterday that a formal Labour decision on transwomen and all-women shortlists, previously said to be a done deal, has been put off until summer.

So never mind Corbyn v the Blairites. That really is Labour fighting the last war, and fighting over petty matters like who gets to run a political party. The war to come is Momentum v the unions, feminists v Brosocialists, and it’s over a question that’s about as fundamental as it gets: what does it mean to be a woman?