Marking the 25h year of the demolition of the 16th- century mosque as a "Black Day", the parties, including the CPI(M) and the CPI, said they would hold mass movements across the country till action was taken against the guilty.
"We will put pressure on the government through mass movements till the people responsible for the demolition of the Babri Masjid are booked," CPI(M) general secretary Sitaram Yechury said at a protest march in Delhi.
"We, six Left parties, take a pledge today that we will not let the country get divided. It is also our warning to the government," Yechury said.
The parties had called for observing the anniversary as a "Black Day" to protest against the "wilful demolition" of the mosque.
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The parties said the Congress, which then ruled at the Centre, and the then BJP government in Uttar Pradesh were both responsible for the demolition on December 6, 1992.
Then UP chief minister Kalyan Singh had given "repeated assurances" to the Supreme Court that nothing would happen to the mosque, she pointed out.
"And yet today Kalyan Singh has been rewarded by being made a governor. It is not a question of an individual. It is the question of an attack by the RSS BJP on the very foundations of the republic of India and the principles of the Constitution," she said at the march.
Thousands of activists on December 6, 1992, had brought down the mosque in Ayodhya, demanding that a Ram temple be built there instead.