'This is Bikram ballet,' choreographer Aaron Cash laughs. We're on the second floor of the Television Ballet building in downtown Havana, here to watch one of Ballet Revolución's daily technique classes, and it is sweltering. Rivulets of human steam are dripping down the turquoise walls, the ceiling fans are static; even Cash – an Australian – has nicknamed the room 'the sweat box'.

And yet, despite an atmosphere that could wilt a cactus, the dancers are slicing the air with crisp turns and neat fouettés, immaculately keeping formation to Ballet Mistress Isis Schery Ramirez's instructions. As they switch and swap groupings, Cash points out various members of the company. Some of the women are en pointe, such as ballerina Barbara Patterson Sánchez. But here also is Lianett Rodriguez Gonzalez, a contemporary dancer and trained gymnast; there is Yasset Roldan Garciarena, classically trained but also a masterful body popper.

It might be the ballet stars that are shining in this particular class but the tables will turn another day when it's time for contemporary, hip hop or folkloric practice. The company – whose previous show, simply called Ballet Revolución toured globally – prides itself on recruiting dancers from a variety of disciplines; not only to showcase different talents but to stretch every dancer's movement vocabulary. All the cast must take classes in everything, including tai chi and yoga, keeping them – apologies – on their toes. It's this fusion of styles combined with Cuban classical discipline that the company is hoping will set audiences alight when it brings new show Balletronic to the Pleasance in Edinburgh this Fringe.

'In Balletronic you learn from everybody in the company,' says Patterson Sánchez. 'Every day you learn something new and that's a challenge for us.' Classically trained at Havana's Escuela Nacional de Ballet (part of the Escuela Nacional de Arte or ENA) Sánchez completed a two-year stint in the Ballet Nacional de Cuba before hearing about auditions for Ballet Revolución. 'Whether classical or contemporary, everyone has their own style of movement. In this show I have to dance contemporary and neo-classical – different to what I'm used to.'

Balletronic rehearsal in Cuba / credit: Alejandro Ernesto

Classical fusion may not seem like a revolutionary idea to some audiences, but in Cuba, Sánchez asserts it is not par for the course. Here tradition rules and the training is fierce. Children as young as eight who show prodigious promise are plucked from the provinces to come to the prestigious ENA and take one of two routes; either contemporary or classical. Those taking the latter path may hope to end up in Cuba's national ballet, a company with a sterling reputation for its dancers' technique, but also one in which tradition is rooted in iron, having been headed by the same director, Alicia Alonso, since 1959. Now 93, Alonso holds a formidable name as a grande dame of dance and in 2010 told The Guardian 'I'll still be running this company in a hundred years' time.'

Contemporary dancer Lianett Rodriguez Gonzalez says that when Ballet Revolución performed their original show in Cienfuegos on the south coast of Cuba it was 'an incidente total' – a real event. 'It was an inspiration for the dancers who came to see it because it was something new. For me, the interesting thing has been the ability to change styles, to go wherever they want you to go. To not be a linear dancer. I like the transformation.'

For Balletronic – as with Ballet Revolución – Cuban choreographer Roclan Gonzalez Chavez is teaming up with Cash, whose eclectic background leads him to happily call himself a 'mongrel' of dance. Cash trained classically but is equally proud of his commercial success, having worked with Twyla Tharp, Cher and LA Ballet, as well as being one of the original Tap Dogs.

'I love coming here because the dancers are so creative,' says Cash. 'Their individuality, their passion, it's that laid-back Caribbean thing. I mean you've hung out with these guys, you can see it.' It's true that in almost every conversation I've had since arriving the theme has been the proliferation of dance and music in Cuban culture. Right now the dancers are chilling out at a barbecue by the seaside but on the way here, onboard a battered off-duty school bus, following a gruelling morning's rehearsal in 'the sweat box', some of them took to the aisles to groove to the bus radio's reggaeton, ignoring the pothole thumps coming hard and fast through the bus floor. The musicians, too tell me that they jam together, sometimes for hours after each rehearsal.

'If you have an idea and they are all into it, it goes even further than you thought it could because everyone gets on board and you can feel it. It's a very open environment – they are incredibly generous with what they can do,' says Cash.

He talks with passion about the individuality of each dancer and remembers with vivid precision their auditions; you sense spending time with the group that there is an immense amount of camaraderie here. But Roclan Gonzalez Chavez has a different opinion on why they work so well in the studio. 'Cuban dancers can give you what you're asking for. The ENA provides very good dancers, not only for Ballet Revolución, but for every show in Cuba – every single style. Ballet Revolución is not the only show in Cuba.' He says it with a smile but it's hard to blame him for being pointed. The notion of Cuban dance carries its own cultural baggage, broadly divided into perceptions of the rigid Soviet-style discipline of the Ballet Nacional or, more commonly – including in Balletronic's own press release – descriptions like 'sizzling' 'raw and 'hot-blooded'.

Balletronic at the Pleasance / credit: Anna Bruce

Chavez is keen to show a rounded but no less popular side. 'Our dancers can do whatever you want. I think that's the case if you're born into a world full of music and full of dance. We lose all our frustration when we dance and when we listen to music – it's food for us. It provides good energy.'

Earlier, back in the studio, he showed us a segment he had been working on for Balletronic. Elegant with clean-lined duets and shifting formations, the piece also played around with narratives of relationships, and the kind of mixture of literal and figurative movement usually found in classical ballet.

But there were other, more specifically Cuban influences at work, too. The kaleidoscopic way the large groups of dancers on stage are managed is typical, co-producer Jon Lee tells me, of the classic 1950s Cuban Cabaret style. Meanwhile Chavez says some of the piece's folkloric moves are associated with the Yoruba gods of Cuba's Santería religion.

'Though rather than folkloric I'd really like to say Afro-Cuban,' he says. 'I think the Africans have a very strong culture and every single place in the world they brought something from their own lives and put it into that area.' There are some 360 African rhythms, Chavez says, the result being that Cuban dancers are 'like a drum – they take the rhythm very fast, they can separate the torso from the leg from the arm from the head.'

Chavez – who earlier admitted one of his childhood stabs at choreography was recreating the moves from Dirty Dancing – doesn't separate these influences out when he choreographs, but just allows his ideas to go where they go. So what is he hoping Edinburgh audiences will make of this? 'They are going to be able to see our country in a proper way. That's what I'm looking forward to.'

Balletronic, Pleasance Grand, 556 6550, until 31 Aug (not 18), 9.30pm, £14.50–£16.50 (£12–£14).