Constellations are determined by the date of the solar calendar. The solar calendar date of June 6th 1976 is July 2nd, and the constellation of that day is Cancer.
Lunar calendar: the sixth day of June, 76.
Gregorian calendar:1Friday, July 2, 976 Cancer
Main branch: Chen Bing, Wu Jia and Mao Yi.
Source: Hua.com.