There's an old saying among Baby Boomers: most of The 60s happened in the 70s.
The aesthetic that you associate with the 60s really didn't emerge until very late in the decade (roughly 1967-69) and carried over into the early 70s. The high-modernist aesthetic of the early 60s has a lot more in common with the late 50s. Similarly, the early 50s have a lot more in common with the mid-to-late 40s than with the later part of the decade.
To put it simply, WWII established a very utilitarian and conformist social aesthetic that began to break down with the emergence of the counterculture in the late 60s. The counterculture itself ran out of steam by the mid-to-late 70s, when all the hippies were turning into yuppies. By that time, a new aesthetic centered around consumerism and technology began to emerge, what we now recognize as the 80s aesthetic.
I see the "60's" culturally starting about 1963-64 with the escalation of the Vietnam War, the publishing of the Silent Spring, and the Feminine Mystique. Then, really peaking probably at Woodstock in 1969.
Of course, no knew it was the peak, so the cultural concepts and trends really continued until around 1975 because the Vietnam War ended, the introduction of the all-volunteer military, the fractionization of the feminist movement, and the passing of several major environmental legislation.
All of these factors caused the ending of the cultural cohesion of the "hippy" or counter cultural movement.
So, what was left over? Drug/party (disco) culture which escalated into cocaine, which we associate with the "80's" really took off in the late 1970's.
Really, to answer your question, the years 1960, 1970 and 1980 don't fit neatly into cultural trends we associate with those decades. I would say a better fit culturally would be 1952-1963, 1964-1975, 1976-1983, 1984-1991, 1992-1995, 1996-2001, 2001-2007, 2008-now.
But these are just arbitrary dates I came up with.
Thinking in terms of decades can be misleading. The cultural narrative which played out in that time was dynamic and complex.
I think you're associating the entire decades with movements which occupied only parts of them. When considering this time period, i find it helpful to ask was it pre-/during/post-Vietnam, and who was in the white house at a particular time. That will help orient you in terms of cultural and political happenings much more so than thinking in terms of decades.
As others has said, cultural trends don't neatly fit in temporal boundaries - there is no causal reason why they should. You also have to view that many different trends happened concurrently with different start and end points.
The media also has a habit of pigeon-holing times. For instance late 60s Britain is seen as the age of the hippy, ignoring the suedeheads/ skinheads that were popular amongst the working class youth.
Also I would like to know what you see so drastically different in the 80s than the 70s? The big hair and make up look that is could be said to define the 80s is really glam rock mixed with some punk and soul sensibilities - a heritage of the 70s.