Two of the Edinburgh International Festival’s main dance events were running concurrently last weekend. That they near filled the city’s two biggest proscenium theatres at the same time – the Royal Ballet of Flanders at the Festival Theatre, Scottish Ballet at the Playhouse – spoke for their drawing power. On artistic results, in my view, Scotland won, but I’ll start with the visiting Belgian team.

Its attractions: a large troupe with a royal name; a full-evening, three-section work seen here for the first time in its entirety in the UK; an imposing title, Impressing the Czar(which, to jump the gun, is as baffling at the end as it might have been enticing when you arrived); and a choreographer, William Forsythe, known to most who have any acquaintance with international dance.

So, was it worth it? The Flanders dancers give their all to Forsythe’s extravagant demands and self-indulgences. But clutter yourself up with the programme’s erudite references to a phantasmagoria of the development of dance and (no less) civilisation, from a random amalgam of symbols from Renaissance art to the tacky trappings of consumerism, and see where it gets you. It got me, none the wiser, out onto Nicol-son Street, flowing like a river in a torrential downpour, which put the tin hat on it.

This piece proposed characters named (and who knows why?) Agnes and Rodger Wilcot, who maintain an increasingly tiresome dialogue on the microphone – unfathomable, like everything else – focusing on someone called Mr Pnut (peanut) and the recurrent image of St Sebastian. The maelstrom of simultaneous business, scurrying around with gilded stage props and bits of paintings (you name it, you may notice it) in pseudo-baroque costumes, occupies some chopped-up “treated” Beethoven with raucous musical eruptions by others; but in and among are taut bits of dance per se that foreshadow the central section, In the Middle, Somewhat Elevated. It was around this, created originally as a stand-alone piece for Paris and familiar from our own Royal Ballet’s repertory, that Forsythe made his unnecessary accretions for Impressing the Czar. It continues to stand alone as a statement (set to Thom Willems’s motorway pile-up score of splintery explosions) of Forsythe’s classical-deconstruction vocabulary: sharp articulations, spearing lines, off-kilter poise. I don’t go for the casual-arrogant walks and insolent air, but this is fierce, “exposed” dance, and the Belgian lead dancers – Aki Saito, Claire Pascal and Wim Vanlessen – give good value in the exposure.

At the climax, the whole company, dressed as schoolgirls, jive and bang around in circles, with nods (but why?) to catchy Giselle, Les Noces and (in the male solos) La Sylphide in kilts. Well, at least those are allusions that make sense, and a mass knees-up is a safe way to end.

Ashley Page’s triumphant transformation of Scottish Ballet has earned its third successive festival showing. Here we have a challenging modern triple bill, including the premiere of a festival commission, Ride the Beast, by the American choreographer Stephen Petronio.

The rasping songs and racketing music of Radiohead left me longing for less, but it whips up a frenzy in high-energy dancing that the audience cheered to the rafters. There is an almost constant flux of movement for the cast, dressed by Benjamin Cho in ribbons and fronds, with tails like exotic birds. With flickery solos, leaping forays and clean, stretched arabesques, they are an ever-shifting mass. It’s theatrically tricked out with light play; you look in vain for any sense that dancers “relate”, but the effect of haphazardness is founded on control. Physically, it’s a knockout.

The veteran American postmodernist Trisha Brown’s MG: The Movieshows discipline in another form: dance rarefied. Take a step backwards, as the soloist Martina Forioso does in variations on her endless circling of the stage, to open the piece at length. Others enter very slowly, as if from another time zone; movement images suggest Eadweard Muybridge’s classic photo-strips of runners in arrested motion. Cool coalescings evolve, but all is mesmeric; a dream world working with and against Alvin Curran’s score, which mixes Satie-esque piano with the voice of John Cage and “found” sounds such as lawn mowers, the foghorn of the Nantucket light-ship, kicked tin cans and the cawing of crows.

After this, it was a pity that a concerted slow hand-clap was provoked by the delay in starting Page’s own tour de force, Fearful Symmetries (caused, doubtless, by the installation of Antony McDonald’s vibrantly elaborate, geometric outer-space decor). This work, made originally for the Royal Ballet, new now to Scottish, is Page’s best: grippingly shaped, pulsed and paced to match the drive of John Adams’s music in its surging ensembles, slick groupings or complex, concentrated duets. Sophie Martin, Claire Robertson and Eve Mutso are three “muses”, perhaps, to the impressive Erik Cavallari, in the athletic central role created by Mukhamedov. The whole cast was splendid; yet another feather in Scottish Ballet’s cap.