To answer this question, we must first look at why the ancient Hellenic calendar was 'lunisolar', meaning based on both the Sun and the Moon. From the civic calendar which has synodic months ('synodic' meaning lunar, from the ancient Greek 'σὺν ὁδῴ', or 'sun hodō', meaning 'with the way [of the Sun]'), we know that the basic calendar was lunar. There is an issue with the lunar calendar, however: The Moon revolves around the Earth faster than the Earth revolves around the Sun. In the time it takes the Earth to circle the sun once, the Moon has circled the Earth 12.36827 times (so between 12 and 13 times). The fact that the two do not match up completely means that the tropical year is 365.25 days, and the lunar year (based on twelve cycles of the 29.53 day synodic month) is 354.36 days. As such, the ancient Hellenes would have 'lost' 10.89 days every tropical year.
The festival year is based upon the lunar calendar. If 10.89 days are lost every tropical year, it will only take a few years for the festival calendar to lag behind the tropical year in such a way that harvest festivals would be celebrated while the seeds are just going into the ground. To prevent this from happening, the ancient Hellenes devised a schedule where an extra month was inserted every few years. We will get to the calculations of that in a little bit. This extra month allowed the lunar calendar to be 'reset' in such a way that the festivals always roughly matched the events they were celebrating. To do this, the ancient Hellenes looked to the Sun, and marked four points in the year where the lunar calendar had to match up roughly with our cycle around it: the solstices and equinoxes.
A solstice is an astronomical event that occurs twice each year (around 21 June and 21 December) as the Sun reaches its highest or lowest excursion relative to the celestial equator on the celestial sphere. An equinox occurs twice a year as well (around 20 March and 22 September), when the plane of the Earth's equator passes the center of the Sun. At this time the tilt of the Earth's axis is inclined neither away from nor towards the Sun. In essence, during an equinox, the period of time the sun is down (night time) and the sun is up (daytime) is roughly the same. The ancient Hellenes observed these four points in the year, and because of that, the ancient Hellenic calendar is partly solar: the solstices and equinoxes are anchor points for the otherwise lunar calendar.
Depending on the city-state, one of these four points was picked for the start of the new year. Athens and Delphi had the summer solstice, Boeotia had the winter solstice, and Milet started out with the autumnal equinox, but moved the new year to the spring equinox around the end of the 4th century BC. This anchor point was the most important; the rest were used to check the accuracy of the calculations.
I addressed the extra month that was inserted to match up the lunar calendar with the tropical one a few paragraphs back. In general--at least from what little evidence survives--we can tell that the extra month was usually inserted roughly half way throughout the year. In Athens, for example, it was usually the month of Poseideon that was repeated. The month became known as 'Second Poseideon', and would have most likely repeated the Mên kata Theion, but not the festivals of Poseideon.
In classical Hellas, an eight-year cycle called ‘oktaeteris’ was known. History seems to indicate that this calendar started off at 776 BC, at the start of the Olympic Games. It approximated the length of the tropical year with (365.25 days) and the lunar year (i.e. 12 synodic months = 354.36 days) with each other by multiplying both by eight. The tropical cycle then had 2922 days, the lunar cycle 2832. The difference between those two lengths is a well-measured 90 days, or three 30-day lunar months. So in a period of eight years, a 30-day month would have to be intercalated three times to reconcile the lengths of lunar and tropical year. In general, this was done in the third, the fifth, and the eighth year of the cycle. Doing this would lead to a calendar of 2923.5 days, or 99 synodic months. Later, the ancient Hellenes measured the oktaeteris as two four-year periods, one of 49 months and one of 50 months, and they called these four year periods 'Olympiads', to accommodate the cycle of the Panhellenic Games.
The Olympic Games were a moveable festival which combines the lunar and solar calendars. The date for celebrating the Olympics was determined as the eighth full Moon following the first full Moon after the winter solstice, as Elis--the city-state where the Olympic Games were hosted--started the year with the winter solstice. The calculation is as follows: start with the first full moon after the winter solstice, then count forward eight more full moons. With this calculation, the first Olympics occurred in (what is now called) August, 776 BC. From this date, count 49 more full moons to July, 772 BC, for the next Olympics, and then count 50 more full moons to July, 768 BC for the third Olympics. This completes the cycle of 2923.5 days, or 99 synodic months every eight tropical years.
By the 5th century BC, the ancient Hellenes realized that 99 lunar months was not a totally accurate equivalent of eight solar years, and that a longer cycle of 19 years would work even better because it was nearly equal to 235 lunar cycles. 19 years x 365.25 days = 6939.75 days, and 235 synodic months x 29.53 days = 6939.55 days. This more precise cycle is attributed to Hellenic astronomer, mathematician and engineer Meton (Μέτων). Meton's cycle assumes 19 tropical years to have 6940 days, as well as 235 synodic months. Seven years of the 19-year cycle would have to have 13 months, the other years 12 months. Where the eighth year cycle ended up off by two days at the end, the cycle of Meton was only off by about half an hour at the end. It was still not completely accurate, though, so Hellenic astronomer and mathematician Kallipos (Κάλλιπος) multiplied the Metonic cycle by four and removed one day so that 76 years had 27759 days. His calculations led him to a cycle that matched up so well, there was only a 22 second difference in favor of the tropical calendar. Interestingly, these new cycles never seemed to have been used for the civil calendars in ancient Hellas, perhaps because they took a boatload of calculations, and were not easy to fit into the now established four-year cycle of the Panhellenic Games.
This concludes the math session for today. Hopefully, this post has cleared up some of the confusion about the importance of solstices and equinoxes. In short: they mattered a great deal for the festival calendar, but they were not celebrated as festivals themselves. All they did was help anchor the lunar calendar to the solar one; an important feat, but not a religious one. Understanding the calendar of the ancient Hellenes matters; it helps bring perspective on the seasonal cycles the ancient Hellenes observed, and helps you realize how in tune with nature they actually lived while always striving to push their society forward by trying to comprehend and work with the natural cycles. They did not take them for granted: they considered how the world worked, and their understanding of the way the Moon revolves around the Earth and the way the Earth revolves around the Sun is a prime example of that.
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