Constantine's play at Lord's Cricket Ground, London, in June 1928 first made British audiences aware of the high quality of West Indian cricket. In the same year, Constantine became the first West Indian player in England to achieve the double of 1,000 runs and 100 wickets in a single season. He was a powerful hitter and one of the greatest fielders of all time.

A solicitor's clerk as a young man, Constantine was called to the Trinidadian bar in 1955 and the next year was appointed minister of works and transport in Trinidad in the new People's National Movement government. From 1962 (when Trinidad and Tobago became an independent nation within the Commonwealth) until 1964, he served in London as his country's high commissioner. In 1966 he became a member of the British national Race Relations Board, and from 1968 he was rector of the University of St. Andrews (Fife) and a governor (director) of the British Broadcasting Corporation. Constantine wrote several books on cricket and one book, Colour Bar (1954), on the race problem. Knighted in 1962, he was created a life peer in 1969.