Is there an orbital significance to where the analemma intersects? Where in the earth's orbit does the intersection occur? I know the general dates when this happens but would like to know if it occurs at a significant position like the perihelion or aphelion.
The intersection corresponds to mid April and late August, the exact date will depend on where we are in the leap year cycle.
Sadly, there's no interesting link between a point in Earth's orbit, nor seasons, and this intersection point.
The analemma can be broken into two parts, the North-South movement, and the East-West movement. The NS movement is due to the change in the declination of the Sun's Geographic Position. And the East-West movement is caused by the Equation of Time. And, in fact, a plot of the equation of time vs declination produce the analemma:
The intersection isn't where either is a specific value, it's just where the two functions happen to have values the same as they had at some other point in the year. They're not equal to each other, not zero, not the start of a season, not half way between seasons, nothing really interesting.
Both functions are a combined function of the Earth's axial tilt, and the Earth's orbit. Plotted above, the EoT is the X coordinate, and the declination the Y coordinate.
An in depth exploration is Demystifying the Analemma, and the Analemma Calculator allows you to explore how changing the Earth's orbit and axial tilt change the shape of the analemma.