The Alberta Serious Incident Response Team issued a news release saying the shooting happened "in an open, highly populated, busy downtown location on a Saturday morning."
The agency is asking anyone who saw or heard anything unusual in the area, or saw any part of the incident, to contact them.
Witnesses have told local media that the man walked up to the demonstration and shouted "white power."
Calgary police officers were nearby, and police have said the man walked quickly away when officers tried to approach him.
The man was confronted by an officer in an alley, and was shot when police say he fought and pulled out a metal pipe.
A metal pipe is shown marked as evidence at the scene of a police shooting in Calgary in this Saturday, March 21, 2015 handout photo. (Government of Alberta - Alberta Serious Incident Response Team)The provincial agency said Sunday that the man remained in critical condition in an intensive care unit.
"The officer caught up with the man in the alley and there was a brief physical confrontation between the man and the officer," the release from the provincial investigative team states.
"At some point, the officer drew his police defensive baton, which was recovered on scene. The man produced a 14 to 16-inch-long metal pipe from somewhere on his person, which was also recovered on scene,"the release continues.
"The officer drew his service pistol and fired, striking the man."
A photo of the pipe that accompanied the release shows what appears to be a bicycle seat post.
Police have said that one officer suffered injuries and was treated at the scene.
The police watchdog has now taken over the investigation.
The release said that police were at the rally to monitor it. It said the man "appeared to be taunting participants" and that verbal exchanges between him and the participants drew the attention of the officers.
When the officers approached the man and he left, they radioed his description to their dispatch, which then sent it out to other officers.
It said a nearby officer in a police vehicle with three police recruits spotted him and attempted to intercept him by pulling in front of him. But it said the man stepped around the vehicle and into an alley.
The officer followed on foot with the recruits some distance back, the release said, before the confrontation happened.
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