Pussy Riot might put the punk rock in modern protest, but their musical chops have always been a little less clear. You will likely know that two of the Russian feminist disrupters were arrested and sentenced to nearly two years in a labour camp for performing their “punk prayer” performance in an orthodox cathedral in Moscow in 2012, drawing international outrage and attention on how Russia treats its dissident artists. But can you remember the serrated riot grrrl riffage of the song that soundtracked it, Mother of God, Drive Putin Away? Or, for that matter, any of their other classics, such as Putin Zassal (roughly, “Putin Has Pissed Himself”)?

Also uncertain is what you’re going to get when you see Pussy Riot billed anywhere. Or, rather, who you’re going to get. Their cast is seemingly interchangeable, like a balaclava-ed Sugababes, with the two global faces of the group staging contrasting endeavours at separate times, of varying artistic quality. The first Pussy Riot tour in North America this spring featured one of those faces, Nadezhda Tolokonnikova, and band playing some of their new anti-Putin agit-electro. At the Edinburgh fringe, meanwhile, the collective’s other leading rioter, Maria “Masha” Alyokhina, is presenting Riot Days, a mix of music, documentary footage and scathing political commentary based on her memoir of the same name.

The one constant is that, no matter how embedded Pussy Riot have become in popular culture, with Tolokonnikova and Alyokhina even appearing at one point in an episode of Netflix series House of Cards, their political actions remain as vital as ever. As long as Putin is in power, their campaign for social justice will never go out of fashion. And so their political pranks have continued apace, still bringing grave repercussions. Last month, memorably, members of the collective, including Tolokonnikova’s husband, skipped on to the pitch during the World Cup final dressed as police officers and high-fived the players. As punishment, they were sentenced to 15 days in prison. And in February, Pussy Riot tweeted that, in Crimea, two of their members had briefly “disappeared” after being detained by Russian security forces.

Alyokhina’s show is poignant, because she very nearly didn’t make it to Edinburgh. Last week it was reported that she had “smuggled herself” out of Russia. She had refused to undertake the community service given to her after she had participated in yet another “unauthorised” protest, and as a result the government had forbidden her to leave the country. In the end she ignored their orders, drove all the way to Lithuania via Belarus, and boarded a flight there instead. And thank god she did, because Riot Days is more than just a gig – it’s somewhere between a gripping piece of Putin-skewering musical theatre, an urgent jazz-punk book recital and a film screening that unfurls like a nerve-shredding thriller.

Two screens show footage of Pussy Riot’s various performances and protests with English subtitles as Alyokhina, in her trademark pink balaclava, recounts every shocking stage of her mistreatment, in Russian, at unrelenting speed. As she whips off her hood and fixes her unwavering stare on the audience, she looks like 1920s muse Marchesa Luisa Casati, training the Kremlin in her sights.

She is joined by three musicians: a saxophonist in sunglasses, a leather-jacketed hype man, and a bare-chested, one-man rhythm section, who have the combined theatrical air of Bauhaus, The Lost Boys and The Blues Brothers. Their music veers unrelentingly between skronky sax solos, ravey techno-punk and even, during the courtroom trial section, Ethio-jazz in the style of Hailu Mergia. It’s an engaging, experimental mix, far more so than the minimalist electronica that Pussy Riot have been making in the US with the LA studio whiz David Sitek.

Sloganeering such as “anyone can be Pussy Riot” and “revolution requires a big screen” flashes in bold lettering as they shout in unison. But while these moments are real fist-raisers, Riot Days isn’t just about punk platitudes. It’s surprisingly funny, as when Alyokhina announces how, when Pussy Riotwent incognito to hide from the police, “we ate whatever God sent our way, which was usually pasta”. And how they knew they were being watched in cafes because “agents wear shoes with pointy toes – that’s their idea of fashion”. They refuse a body cavity search on entering prison because “if you bend over, you’ll be bending over for the rest of your life”.

There are flashes of vulnerability and self-doubt amid all the chest-beating too. “Do I have the right to do this or am I a barbarian?” she recalls asking, before Pussy Riot took to the church altar to dance. An image of her toddler son holding a sign saying “let mom go” during her prison sentence packs a sombre punch.

Ordinarily, it would be hard to maintain attentiveness during a show with this much monotonous shouting and no pause (at one point, someone in the crowd turns to another and asks: “When do you think they’ll start singing?”). But it’s impossible to look away, even when Alyokhina’s joint vocalist, who dances like Jason Orange by way of Henry Rollins, sprays the crowd with water. After Riot Days’ short tour in the UK, what will happen to Alyokhina when and if she goes back to Russia appears to hang in the balance. And yet in the show’s clever conclusion, she suggests that, no matter where she is, whether she’s incarcerated or abroad, her dedication to her cause and her impassioned independence will always mean that she is free. The lights go up on the audience and the band intones: “Are you?”