This is in addition to two other complaints against the company.
Police in Delhi filed the case after the National Commission for Protection of Child Rights complained that Twitter was promoting child pornography.
Twitter has been at loggerheads with the Indian government over controversial new IT rules.
The second case, which was filed in the central state of Madhya Pradesh, accuses the company of sowing an allegedly incorrect map of the country. The map, which has now been removed, did not include Jammu and Kashmir, and Ladakh as part of India in the “Tweep Life” section of Twitter.
It was filed against Twitter India chief Manish Maheshwari, who is also facing another investigation in connection with the map in Uttar Pradesh state.
Mr Maheshwari is also facing another complaint in connection to a video showing an assault on a Muslim man in Uttar Pradesh earlier this month. The police denied any religious angle and charged some journalists and politicians, along with Twitter and news website Wire for sharing the video with the “intent to a riot, promoting enmity and criminal conspiracy”.
The investigations against Twitter – four in less than a month – come amid a widening rift between the platform and the federal government over new rules for digital media platforms.
The Information Technology (Intermediary Guidelines and Digital Media Ethics Code) Rules were announced in February and became effective in May.
They require social media firms to appoint three full-time executives – all Indian residents – who will be empowered to remove content when requested by law enforcement and judicial bodies. Platforms such as Facebook, Twitter and Google must also track the originator of a message if asked by a court or the government.
On Tuesday, a parliamentary panel, headed by opposition MP Shashi Tharoor asked Facebook and Google to comply with the new IT rules.
Mr Tharoor also asked Twitter India why his account and that of IT minister Ravi Shankar Prasad were blocked briefly recently following complaints of copyright violations.
It’s rare for social media company executives to be quizzed over posts on their platforms. Under Indian law, these firms are “intermediaries” that cannot be held responsible for content on their sites.
But the government says Twitter could lose its intermediary protection if it does not comply with the new rules.