The motorcycle and the car were burnt by Gorkha Janmukti supporters at Rambi on the National Highway 31 where they were picketing since 6 AM when the shutdown began, police said.
The police intervened and arrested 12 GJM supporters, while the remaining fled from the picketing spot on the arterial highway which connects West Bengal with Sikkim.
Darjeeling town wore a deserted look with traffic off the roads and markets and offices closed.
Superintendent of Police Kunal Agarwal said that stringent action would be taken for destruction of government property and that the police would ensure that the NH 31, Sikkim's lifeline, remain opened.
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District Magistrate Soumitra Mohan also said that strong steps would be taken in case of breakdown of law and order.
The bandh is to demand Gorkhaland, which is a 107-year-old demand, the GJM said.
A thousand GJM supporters were picketing at Chowkbazar with party flags, while large gatherings of party supporters were also reported from Jorbungalow, Mirik, Sukhiapokhri, 9th Mile and 26th mile.
Some foreign tourists still remained in the hills, while around 2000 domestic ones left yesterday, police said.
A few shops opened in the early morning, but closed by 7:30 AMas there was no customer.
The only traffic on the roads were ambulances and vehicles carrying people to weddings or sradh ceremonies.
There was no work in the tea gardens in the hills.