“Nobody has to remain the person they were born; we can put ourselves together like a jigsaw,” says Bujar, the androgynous Albanian narrator of Crossing, the second novel by Finnish-Kosovan writer Pajtim Statovci, acclaimed for his debut My Cat Yugoslavia.

Bujar’s story stress-tests the truth of his statement, asking: at what cost? We see him in Rome in 1998, in hospital after a suicide attempt, and eight years earlier in his native Albania, as a teenage runaway heading for the coast with his schoolmate Agim, who wants to leave after his father beat him for wearing his sister’s clothes.

The boys, whose semi-erotic relationship is built on neither of them asking awkward questions, flog stolen cigarettes in the hope of making enough cash to get across the Adriatic.

We cut from these exploits to watch Bujar criss-cross Europe at the dawn of a new century, starting afresh – and alone – in countries where, as he puts it, you “have time... to be traumatised for years about something utterly trivial”.

His experiences are often grim. In Berlin, he gets a restaurant job only to be raped by the boss. But in Helsinki, he auditions for an X Factor-style show, where he’s nearly booted out without singing a note for being too boring, until he turns his fortunes upside down by tearfully telling the judges he’s a trans woman from Turkey.

There’s comic tension in how that episode plays out, not least because Bujar is stealing his lover’s life story. Another funny moment sees him join a writing course in the guise of a woman from Bosnia, outdoing the woes of other students equally keen to turn pain into prose. Yet part of what makes Crossing unpredictable is how swiftly horror replaces laughter: witness how another student seduces Bujar only to turn violent at what his roving hands discover.

He’s prone to violence himself, punching a lover in Madrid who laughs at the sight of him trying on her clothes

Translated from Finnish by David Hackston, Statovci’s writing is slyly artful. By September 2001, Bujar is in New York, where a drag artist nicks the money he’s saved from 80-hour-weeks of pot washing; 9/11 isn’t mentioned, which only underlines Bujar’s position on the margins. Next to nothing is said about his sister, who goes missing before he leaves Tirana. Bujar later admits he “consciously avoids” the likelihood that she’s fallen victim to people traffickers.

“I am a 22-year-old man… Sometimes I am a 22-year-old woman,” says Bujar. His Christian mother is the daughter of diplomats from Albania’s capital, his Muslim father is of Kosovan farming stock. It’s in keeping with the novel’s studiously non-binary logic that Bujar isn’t a straightforwardly sympathetic narrator. Duplicitous, he’s prone to violence himself, punching a lover in Madrid who laughs at the sight of him trying on her clothes (she had come home from work early).

Added complexity comes from how his time-hopping casts doubt over quite how he got to Italy in the first place. That question keeps you reading even when you feel daunted by his gruelling narrative. The answer is a shock, but one that befits the harsh perspective on show in this cruel odyssey.

• Crossing by Pajtim Statovci (translation by David Hackston) is published by Pushkin Press (£14.99). To order a copy go to guardianbookshop.com or call 0330 333 6846. Free UK p&p over £15, online orders only. Phone orders min p&p of £1.99