Did the former ruling princely families of India continue to hold notable social influence after they were de-recognized in 1971?

Most of them did not. But the Scindia dynasty of Gwalior has continued to be dominant in both of India's major political parties: the Congress and the Bharatiya Janata Party.

It begins with Maharaja Jivajirao's death. The rise of the Scindia family in Indian politics predates 1971. In 1962, his widow, Rajmata Vijaye Raje Scindia, contested elections for the Congress party from the Guna Lok Sabha seat in the state of Madhya Pradesh. Five years later, she contested from their home turf of Gwalior and it has remained a Scindia bastion ever since.

The rise of Indra Gandhi and the left-wing of the Congress was too much for her, however. In 1967, she quit the Congress party and joined the Swatantra Party, which was the only centre-right political party in India's history to become a National party. The party's leader, C. Rajgopalchari, was himself an ex-Congress member and a freedom fighter who had become disillusioned with the Congress and had formed his own party which stood for classical liberalism. C. Rajgopalchari was particularly popular with the erstwhile Hindu rulers and aristocracy as well as vaishya businessmen across North India, despite being a Tamilian himself. This was evident with the best performance of the party being in Rajasthan, Orissa, and Gujarat in the 1967 election. Interestingly, Swatantra party's performance in Rajasthan was linked to the support of Rajput rulers. One in particular was Gayatri Devi, the widow of the Maharaja of Jaipur. This may perhaps have been one of the factors which led to Indira Gandhi's decision in 1971 to remove privy purses for the erstwhile princes.

The Swatantra Party began to decline after C. Rajgopalchari's declining health and ceased to exist after his death. At this point, Scindia joined the Bharatiya Jana Sangh and became a prominent member. She rose up the ranks in the organisation and was a prominent MP when the Jana Sangh merged with the Janata Party. She even contested against Indira Gandhi in Rae Bareli, but lost. She was jailed during the emergency along with Gayatri Devi and shared a cell with her in Delhi's Tihar jail.

She became a major hardliner in the the BJP after it split from the Janata Party and played a major role in the Ram Janmabhoomi movement. Later on, she became a vice-president of the BJP and retained the position until her death in 2001.

Her daughters are staunch members of the BJP. Yashodhara Raje has been prominent in Madhya Pradesh politics, but Vasundhara is the one who became most successful, twice serving as the chief minister of the state of Rajasthan. Her son, Dushyant Singh, also a royal from his father's lineage, is currently an MP for the BJP.

Now, Madhavrao Scindia, Vijaya Raje's son, is another interesting story. He originally contested as a Jana Sangh member at the age of 26 and won her former seat, Guna. However, he soon began to have issues with her in the form of property disputes and broke away. He never joined the Janata party, but he did contest as an independent in the next election. Later on, the complete division of his family led to him joining the Congress party in 1980. He remained a member his entire life. In 1984, he became the minister of railways under the Rajiv Gandhi cabinet. In 1991, under the cabinet of P.V. Narasimha Rao, he became the minister of civil aviation. In 1995, he became the minister of human resource development. There was even a brief phase when he broke away from the Congress party and formed his own party, the Vikas Congress, but that brief rebellion only lasted a few months. He died in a plane crash in 2001 while he was still a sitting MP for the Congress.

Jyotiraditya Scindia, his son, is currently the most prominent royal in Indian politics today, especially after his aunt's electoral defeat in Rajasthan a few months ago. He has been elected from both Guna and Gwalior, and currently holds the latter. He had been a minister of state without portfolio (a junior minister) from 2012 to 2014 under the Manmohan Singh cabinet. Most recently, in the Madhya Pradesh election, he almost became the chief minister—but stepped down in favour of Kamal Nath. Rahul Gandhi then appointed him to lead the Congress' campaign in Uttar Pradesh along with his sister, Priyanka Gandhi.