We tend to refer to murders as whodunnits, says DCI Gareth Morgan, but he believes a better term for the extremely odd case that unfurled under his watch in south Wales during 2015 would be a “who-is-it?”. When a heinous, seeping package – a decaying corpse half-preserved in more than 40 layers of wrapping – is discovered in the village of Beddau, a woman named Leigh Ann Sabine – who recently died of brain cancer – quickly becomes the prime suspect. The problem is, police cannot work out who it is she has killed.

Ultimately, The Body Next Door is not really a who-is-it or even a howdunnit; it is – far more compellingly – a how-could-she-do-it? We first encounter Sabine through the eyes of her neighbours in Beddau (whose pronunciation seems to be a local bone of contention, but the residents featured here insist on “beh-tha” rather than “beh-thai”). They understood her to be a narcissistic fantasist with an obscure past, an affected accent and a penchant for fishnets well into middle age, but reminisce about her eccentricities fondly. By the time this series is over, however, their affection is hard to stomach. For beneath the dense layers of this breathlessly twisty investigation lies a more resounding mystery, as we are forced to reckon with a character whose actions range from heartless to inexplicably vile.

Generally, good true-crime TV requires three things: pace, clarity and some moral boundaries. This addictively watchable three-parter is certainly pacy – the case itself is so convoluted, surprising and sweeping (it traverses the globe, and five decades) that there’s no need for any infuriatingly glacial recapping or other filler. That’s because woven through the police inquest to identify the victim – a complex undertaking in itself – is a retelling of a different, even more jaw-dropping transgression. This one was huge news half a century ago in New Zealand, where Sabine and her husband John had settled and had five children. One day, in 1969, the couple abandoned them all at a daycare centre, returning – temporarily – without explanation over a decade later. There was a national outcry, but Sabine never showed any remorse. Three of the children are interviewed extensively in this series, and their experiences are heartbreaking – two of the girls were sexually abused by a foster father – but also so unremittingly bizarre that even son Steve thinks of the whole ordeal as an absurd nightmare (“I sit there sometimes at home by myself and I can’t believe it’s happened”).

The Body Next Door does stumble slightly when chronicling the early days of the investigation. To be fair, the chain of events that led to the discovery of the corpse is so confusing – it involved a neighbour attempting to play a prank with a medical skeleton – that even after multiple viewings of the documentary and reading detailed reports of the case, I still can’t quite make sense of it. However, from that point on, the show does a sterling job of flitting between 1970s New Zealand, 1980s Reading and 2010s Beddau without ever losing its strong narrative thread.

Ethically, the show has a similar false start. Early on in the investigation, police arrest the neighbour, who found the body (understandable, considering the baffling discovery story). She was later completely exonerated, but this series still shows a fair amount of footage from her police interview, in which she appears extremely distressed. At this point, nobody can plausibly deny that all true-crime TV is essentially exploitative – by definition, the genre mines entertainment from trauma and tragedy – but recently a more compassionate, victim-centric style has emerged (Raw TV, who made this series, have form in this regard with the likes of The Tinder Swindler and American Nightmare). The Body Next Door would probably like to put itself in this category – and after this misstep, the series does cleave to these new rules, giving the Sabine children space to tell their stories, and acting as a thoughtful meditation on many fascinating themes – the elemental horror of maternal neglect; coercive control; generational trauma – that is troubling in a psychological-puzzle way, rather than an uncomfortably queasy one.

Which is not to say that The Body Next Door isn’t utterly nauseating. The foul corpse package, which keeps returning to the screen in grainy horror film-style footage taken from the autopsy lab, becomes an emblem of the rank inhumanity at the heart of this story: not only Sabine’s grim lies and evasions of responsibility, but also the nastiness she had in her, a darkness likely kindled by her own horrific experiences in the care system. A niggling question hovers over this programme about how much Sabine would have revelled in the attention it brought; towards the end of the series, a neighbour reveals that she once boasted “after I die, I will be famous”. But surely nobody – no matter how sociopathic – would want an obituary like this.