Final answer:
The band from the British Invasion that also played a central role in the 1960s counterculture is B) The Beatles, having started in Liverpool and becoming major cultural figures by Woodstock in 1969.
Step-by-step explanation:
The band that was not only a part of the British Invasion but also a significant figure within the counterculture rock 'n' roll movement of the 1960s is B) The Beatles. Originating from Liverpool, The Beatles soon became the embodiment of suburban youth culture in the mid-1960s.
They offered a more middle-class version of the rebellious postures of earlier generations and played a pivotal role in the counterculture movement.
Their popularity was part of a broader cultural shift where rock 'n' roll, initially born from amplifiers, electronics, and traditional rhythm and blues, became a mainstream and unifying force in both the UK and the USA.
By the time the Woodstock festival occurred in 1969, rock 'n' roll had delivered a powerful expression of youth emancipation, with The Beatles being central figures in this era.