The ancient symbol of the double spiral appears to be closely related to the sun's path throughout the year, and to incorporate the subtle complexities created by the obliquity of the ecliptic and the eccentricity of earth's orbit.

In his books The Stars and the Stones: Ancient Art and Astronomy in Ireland, archaeoastronomer and artist Martin Brennan provides compelling evidence that the double spiral is related to the changes in the sun's path throughout the year.

He notes that the shadows cast by a gnomon throughout the year will create a straight line east-west on the equinoxes, but that on the solstices the shadow's path will actually trace out two hyperbolas (this phenomenon is discussed in a previous post here). Brennan explains how this fact may lead to the double spiral design:
At summer solstice the shadows are shortest and the arc is concave. At winter solstice the shadows are longest and the arc is convex. In archaic astronomy, these were known as the 'horns of the solstice.' At equinox the shadow is straight. If the shadows of the sun are correlated over the period of one year in chronological order following their curvature they form a double spiral. In winter the spiral is counter-clockwise and the coils are wide. The shadows begin to straighten as equinox approaches, and after equinox they begin to wind into a clockwise spiral and tighten. They contract until the summer solstice, straighten again at equinox and return to a left-handed spiral again in winter to continue the process perpetually.

The Boyne Valley artists developed the double spiral and displayed it prominently. Recently, an American artist, Charles Ross, arrived at a double spiral in a controlled experiment documenting the sun's path through the year. Using a stationary focused magnifying glass, he placed wooden planks in a fixed position for 366 consecutive days. The sun's rays burned a pattern in the planks which when graphed showed a precisely executed double spiral. 190.
The burned planks on which Charles Ross has performed this experiment can be seen in this photograph on his website, and the spiral pattern can be seen inlaid on the floor of the gallery (click on the third image from the left at the very bottom of that webpage).

Charles Ross has also created three-dimensional solar art called Star Axis in the desert of New Mexico, including a "shadow field" which illustrates the shadow paths throughout the year from one solstice to the other. By visiting this excellent webpage about the project, visitors can select the winter solstice, summer solstice, or equinox position, and then press "play" to see an animation of the shadow movement on those important annual earth-sun positions (to reach the animations, follow the link above and then click "Solar Pyramid and Shadow Field" in the central horizontal menu bar; next click the link which reads "Shadow Field" in the text portion of the page).

This graceful annual solar motion is also related to the analemma pattern created by the earth's tilt, which causes the ecliptic path to move back and forth across the celestial equator throughout the year (crossing at the equinoxes, as discussed in this previous post and elaborated in greater detail in the Mathisen Corollary book). The other phenomenon which causes the analemma's shape to look the way it does is the eccentricity of earth's eliptical orbit, which causes the earth to speed up as it "falls towards" the sun on its way to perihelion around January 4th each year and to slow down as it "rises away" from the sun on its way to aphelion around July 4th each year. Because the earth is moving faster in its orbit on some parts of its orbit, the sun does not "make it" to the anticipated point (for instance, the culmination point or "high noon" point) at the same time on days when the earth is moving faster as it speeds towards perihelion (because it is still spinning at the same rate, an observer on earth will perceive the sun as lagging a little from one day to the next at the same exact time).

The celestial mechanics surrounding the graceful figure-eight shape of the analemma are thoroughly and superlatively explained in the series of pages and animations in this website from Bob Urschel (use the small blue arrows at the bottom of each page to go to the next page -- it will require looking at all of the pages and videos to fully understand this complex process). The video which shows the sun's ecliptic path moving over and under the celestial equator throughout the year, and tracing out the figure-eight analemma as it does so, can be seen here. The first successful photographic record of the analemma, along with arcs showing the sun's path on the solstices and one equinox, can be seen here.

Having established the connection of the double spiral to the graceful annual motion of the sun from one solstice to the other and the equinoxes in between, we can then note the presence of this powerful symbol around the globe. As has already been noted in the citations from Martin Brennan, it can be found in the petroglyphs adorning the megalithic architecture of the Boyne River Valley in Ireland (where the passage mounds have clear alignments to the solstices, equinoxes, and cross-quarter days). It is also found in the New World, such as in the double spiral shape pictured at top which is carved into Fajada Butte in Chaco Canyon, New Mexico.

We have also noted the presence of spiral patterns on the faces of some of the Tocharian mummies of the Tarim Basin. Martin Doutré has made a convincing argument that this very same solar double-spiral pattern found on the faces of some of the Urumqi mummies is directly related to the double-spiral pattern found in many of the mokos or facial tattoos of the Maoris of Aotearoa -- scroll down to the section entitled "Origins of the Early Maori Moko (Facial Tattoo)." For a powerful example of the double spiral, which was often seen crossing the bridge of the nose in men, as well as along the cheekbones and in other areas of the mokos, see this beautiful portrait of Maori Chief Wi Te Wanewha by Gottfried Lindauer (1839 - 1926).

The presence of the double spiral among the ancient inhabitants of Ireland, the Tarim Basin in China, Chaco Canyon in New Mexico, and Aotearoa or New Zealand can of course be explained by independent and isolated observation of the solar patterns, although it must be admitted that this pattern is extremely subtle and not at all inherently obvious from a casual observation of the arcing hyperbolas of the gnomon's shadow field. It is also possible to explain its widespread appearance as a result of ancient trans-oceanic contact and migration. If it were the only data point to support such a theory, it would not be very strong, but taken together with the extensive other data points which exist in mythology and archaeology, it is a noteworthy addition to the debate.

Finally, William Lasseter has some interesting musings about the possible connection of the serpentine double-spiral to the twisting, spiraling dragons which appear in art and tradition the world over, including in Europe in a blog post here. Interestingly enough, that post also includes the page from Martin Brennan's book explaining and illustrating the double spiral that is quoted above in this post, as well as some very insightful literary analysis of J.R.R. Tolkien's The Hobbit. For readers who are interested in Tolkien and the connection to the subject of the celestial phenomena, be sure to check out this post about the connections between the crucially important constellation of Orion and the elven king Earendil.