I don't get it? why they fight?

Why do dogs chase cats?

Because they are different from each other and as humans we fear what is different and what we fear we attack. Its basic gang mentality.

-Mithaearon, deep in The Medway Delta
Remember Cthulhu loves you.

I thought it typifies the human condition:

We understand whites fighting blacks, but are dumbfounded by Protestants v Catholic wars in Northern Ireland, or Sunni v *beep* or Hutu v Tutsie, Hindu v Moslem.

As I believe maybe Kurt Vonnegut wrote: "if all forms of discrimation were gone one morning, there would be new ones by noon"

By the way: vigilante Christians beating up gays from 1970-2010 in the USA

It was a tit for tat, escalation thing. Not productive, I agree. However, they did not listen to the same music. The rockers, who were more rural and working class, were still holding out for Elvis type rock; the mods, who were from the urban middle class, were into the latest and hippest dance hall sounds and kind of dressing up.

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See a list of my favourite films here: http://www.flickchart.com/slackerinc

Ancient (pre-Christian) British tribal rivalries that re-emergence and imprint themselves on new generations of youth.

I watched a documentary a while ago that suggested that a lot of the tension was stirred up by the media.
Easter weekend (which, for non-UK readers, is a 4 day weekend here) of 1964, a load of youths went to Clacton on Sea (a beach resort in Essex), some groups of "mods" and some groups of "rockers". Problem was, Easter was early that year, and the wintry weather was still going on. Consequently, a lot of the seasonal places that these kids would normally go to were still closed.
With nothing to do but wander the freezing streets, some of the visitors began committing acts of vandalism/criminal damage, couple of smashed windows etc as would happen in most city centres at the weekend, and a few dozen of them ran onto the pier which was still closed for the season - not that there was anything to break into there because there were no shows or anything on, what with it being closed! They were just doing it for a lark, as kids do - and this prompted the guy guarding the pier to call the police, who were ready to arrest them when they were done.
All in all, though, no group-related violence (or fighting of any kind) took place.
The newspapers, however, exaggerated the extent of vandalism/criminal damage that had taken place to say that the youths had been rioting, and worse still, wrongly stated that the mods and rockers had been fighting against each other.
The sensationalist headlines whipped up a frenzy, it made mods and rockers a national thing (instead of just London based and a cult/niche lifestyle), that everyone under the age of 25 had to pick a side etc
It meant that the next holiday weekend, both "sides" hit the coastal areas in numbers to "protect their own" against a potential onslaught by the other side. Both sides were on the defensive and also fired up on booze and/or drugs. So the large scale riots that happened were like a self fulfilling prophesy because such a big deal had been made out of some very minor disorder in Clacton a few weeks earlier that wasn't even anything to do with any mods/rockers rivalry.