* Both sides step up tactics in two-month-long protests
* Police fired tear gas in underground stations at weekend
* Police have never used water cannons since purchase in 2014
* Demonstrations have roiled Hong Kong's summer (Recasts)
HONG KONG, Aug 12 (Reuters) - Hong Kong police put on a demonstration of an anti-riot water cannon on Monday, in a warning to protesters as authorities toughen their approach over violence that has roiled the Asian financial hub for two months.
This summer's increasingly violent demonstrations have plunged the Chinese-ruled territory into its most serious crisis in decades, presenting one of the biggest popular challenges to Chinese leader Xi Jinping since he came to power in 2012.
Police have never used the cannon since two were bought in the wake of pro-democracy protests in 2014, but Monday's demonstration used one to blast at dummy targets at a training facility as tactics on both sides harden and shift.
The police were condemned for heavy-handedness at a news conference by three individuals who said they represented the protesters and reiterated a demand for an independent panel to investigate incidents of excessive force.
"It's not just the water cannon," said one of the three, Steven Ng, who wore a mask. "The police are continuing to use all sorts of weapons to challenge the bottom line of Hong Kong people with their weapons...All Hong Kong people with a conscience can see this clearly."
Over the weekend, as demonstrators threw up barricades across the city, police shot volleys of tear gas into crowded underground train stations for the first time, and fired bean-bag rounds at close range.
Scores of protesters were arrested, sometimes after being beaten with batons and bloodied by police. Police have arrested more than 600 people since the unrest began more than two months ago.
Tear gas was fired at the blackshirted crowds in districts on Hong Kong island, Kowloon and the New Territories, with one young female medic hospitalised after being shot in the right eye, triggering a protest by medical workers who wore bloodied patches over their eyes.
The protests began in opposition to a bill allowing extradition to the mainland but have widened to highlight other grievances, drawing broad support.
Monday's demonstration, which showed the water cannon was powerful enough to force dummy targets back at distances of 30 to 40 metres (33 to 44 yards), drew a rebuke from global rights group Amnesty International.
"Water cannons are not a toy for the Hong Kong police to deploy as a sign of strength," Man-kei Tam, the group's Hong Kong director, said in a statement.
"These are powerful weapons that are inherently indiscriminate and have the potential of causing serious injury and even death."
But some establishment voices called for its deployment.
"You're gonna taste the water cannons! Very soon!" a pro-Beijing lawmaker, Junius Ho, who attended the demonstration, wrote on his official Facebook account.
As police have dialled up their aggression, protesters have sought to channel a Bruce Lee maxim: "Be water," making use of a flash-mob strategy to frustrate authorities and stretch their resources.
For a fourth straight day on Monday, thousands of activists peacefully occupied the arrivals hall at Hong Kong's busy airport, shouting, "No rioters, only tyranny!"
An online fundraising appeal garnered $1 million in roughly an hour and more protests were planned near the city centre.
Demonstrators say they are fighting the erosion of the "one country, two systems" arrangement enshrining some autonomy for Hong Kong when China took it back from Britain in 1997.
They are demanding the resignation of the city's leader, Carrie Lam, and an independent investigation into the handling of the protests.
Authorities have called the demonstrations illegal and dangerous, while highlighting their impact on the already-faltering economy and residents' daily lives.
Beijing says criminals and agitators are stirring violence, encouraged by "interfering" foreign powers including Britain.
China is also putting pressure on big companies, such as Cathay Pacific Airways, whose shares tumbled to close to a 10-year low on Monday, after it was told to suspend staff engaged in illegal protests.
North Korea said late on Sunday it fully supports China on the situation in Hong Kong, which it also blamed on "foreign forces" interfering in an internal affair of China to encroach on security and order.
China is the main diplomatic ally and economic benefactor of isolated North Korea, which has been under international sanctions for its nuclear and ballistic missile development. (Writing by Farah Master and Tom Westbrook; Editing by James Pomfret and Clarence Fernandez)