Start and end dates vary with location and year. Since 1996 European Summer Time has been observed from the last Sunday in March to the last Sunday in October; previously the rules were not uniform across the European Union. Starting in 2007, most of the United States and Canada observe DST from the second Sunday in March to the first Sunday in November, almost two-thirds of the year . The 2007 US change was part of the Energy Policy Act of 2005; previously, from 1987 through 2006, the start and end dates were the first Sunday in April and the last Sunday in October, and Congress retains the right to go back to the previous dates now that an energy-consumption study has been done.
Beginning and ending dates are the reverse in the southern hemisphere. For example, mainland Chile observes DST from the second Saturday in October to the second Saturday in March, with transitions at 24:00 local time. The time difference between the United Kingdom and mainland Chile may therefore be five hours during the Northern summer, three hours during the Northern winter and four hours a few weeks per year because of mismatch of changing dates.
The United Kingdom stayed on daylight saving time from 1968 to 1971.
--- So it looks like the symmetry is between the Northern and Southern hemispheres rather than between the two parts of the year.
I am senitmental about the UK remaining on GMT for some of the year, but that's probbaly a sentiment better reserved for History Club!