There's an interesting video on the Follow-Rugby.com website that traces the evolution of the All Blacks' haka, demonstrating that the haka as we know it today has undergone some very significant changes over the past century.
However, the brief clip it shows of the 1922 haka is neither the All Blacks nor rugby union. It's actually the 1922 Maori rugby league tourists playing New South Wales Seconds at the Sydney Cricket Ground, a game the Maori lost 31-14. As well as a long take of the haka, the footage is notable for showing an early version of the play-the-ball, which looks like a mini-scrum (as its originators intended).
YouTube also has footage of the haka being performed in 1926 by the New Zealand rugby league tourists to Britain, known at the time as the Professional All Blacks, at the first test match of the tour at Wigan. Oddly enough, it seems that Pathe News could not distinguish between league and union, so the clip is labelled as a rugby union match! The haka is led by Phonse Carroll, the NZ hooker who was a relative of Brisbane Bronco and Leeds Rhino Tonie Carroll and, more importantly, a conscientious objector in World War One.
To say the least, the tour was not a success - you can read the full story in John Coffey's excellent book The Tour That Died of Shame - but the British Pathe clip shows the two sides running out, the haka and some of the play. The first two players out for the British side are captain Jonty Parkin followed by Jim Sullivan. The Kiwis are led out by Mildred Mair (the wife of the tour manager), carrying a NZ flag and a stuffed kiwi, followed by captain Bert Avery.
At the time the match took place, hundreds of thousands of miners were locked out by the pit-owners - 40,000 in the Wigan area alone - and five months previously Britain had been gripped by the General Strike. Despite the hardships that were been inflicted on the local miners, fifteen thousand people turned out to watch an exciting 28-20 win for Britain.