Addressing a gathering at a function to mark the Congress' 133rd Foundation Day, Gandhi hit out at the BJP for weaving a "web of deceit" and said his party would defend the truth even if it suffered or lost elections.
Gandhi, who was appointed party president earlier this month, unfurled the Congress flag at the Congress headquarters in Akbar Road, a first for him. Former prime minister Manmohan Singh and senior leaders, including Congress Working Committee (CWC) members, were present at the event.
"Today, it is distressing to see that this document, the foundation of our country, given to us by the Congress party, given to us by Mr Ambedkar, is under attack.
"... Statements are being made by senior members of the BJP and it is under attack surreptitiously from the back. It is the duty of the Congress party, the duty of every single Indian, to defend the Constitution and every person's rights and views," he said.
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"And this is the difference between us and them," he said.
"We might suffer, we might not do well, we might even lose, but we will not give up the truth, we will defend the truth," Gandhi added.
He also highlighted the Congress' "glorious" history, stating that the party has been working in the interest of the country for over a century.
Gandhi, who distributed sweets to children on the occasion, said the Congress had many achievements to its credit, achieved with the help and together with the Indian people.
The party was "jointly responsible" for bringing independence to the country, he said.
Hegde, the Union minister of state for skill development, reportedly said at a function in Karnataka that people should identify themselves with their religion and "those who, without knowing about their parental blood, call themselves secular, they don't have their own identity..." He also reportedly said, "We are here to change the Constitution and we'll change it.