Near as I can tell, the halfway point between the Spring Equinox and the Summer Solstice (called the Celtic cross quarter of Beltane, though it predates the Celts and goes back to early agrarian cultures) is today, May 5, 2023, at 7:35 PM Pacific Daylight Time.

The Autumn Cross-Quarter occurs 7 November 2014, 11:56 UTC (3:56 Pacific Standard Time). This marks the point on the Earth's circuit around the sun that is half way between the autumn equinox and the winter solstice.

Astronomers of antiquity measured these dates with a high degree of precision, using what we would consider crude instruments and cumbersome mathematics. 

One reason the cross-quarters matter is that it has been the common sense of humanity since these observations began, that the solstices are MID-summer and MID-winter.  June 21 and December 21 (or thereabouts) are not "the first day of summer/winter" as the publishers of calendars would have it. 

The calendar is about our shared experience of length of daylight and how it changes, and how the sun's pattern (years) interacts with the moon's patterns (months).  Like so many things in the modern world, the calendar has been simplified (I admire the self-reliance of the lunar calendar of Judaism!), sanitized, and accepted as a thing in itself.  We forget the calendar is a story about phenomena.

Imbolc (say it "ih-molg") is the astronomical cross-quarter between the Winter Solstice and the Spring Equinox.  In Ireland it is called St. Brigid's day -- St. Brigit of Kildare, that is.

My ancient ancestors divided each golden year into "pieces of eight", as it were.  They marked and celebrated the ongoing life and the cycle of the seasons.  Don't put too much stock in self-styled neo-pagan (read "Victorian") trappings draped on solid archaeo-astronomy. 

In 2013, the spring cross-quarter actually occurs 3 February at 15:57 UTC.  Its actual date varies because of the vagaries our leap-year calendar. 

As a result, celebration of the cross-quarter varies by culture.  In Ireland, St. Brigid's day is the First of February.  In Asia, the new moon following the spring cross-quarter is the start of the new year (e.g., Tet in Vietnam).

Today is a cusp of the seasons, the turning of fall into winter. 

Face it:  if the Summer Solstice is Midsummer Night, and we sing about the Bleak Mid-Winter at Winter Solstice time, then the solstices and equinoxes can not be "The first day of ----" as the calendar publishers would have it.