It is correct that the Earth's axial tilt alone would result in an analemma of a figure of eight with equally sized lobes. The axial tilt contributes to the equation of time.
Circular Orbit, no Axial Tilt
Let's look at an example case with a perfectly circular orbit and no axial tilt, to observe what happens with no equation of time. In this example, it's noon at the equator and prime meridian. The Sun is directly overhead. Next, the Earth rotates exactly once. This isn't a normal solar day, but a sidereal day, about 23 hours, 56 minutes. The Earth has rotated exactly once with respect to the stars. Where is the Sun directly overhead now? The Earth has moved a little bit in it's orbit around the Sun, so that our observer at the equator and 0 degrees longitude has observed that the Sun has moved in the sky with respect to the stars. The Sun is now directly overhead at a location to the east, a little less than 1 degree longitude away (360 degrees in a circle, usually 365 days in a year). The Earth must rotate a little more so that the Sun is directly overhead again (which completes the 24 hours in a day).
After each rotation, the Earth must rotate a little more each day to bring the Sun back overhead at 0 degrees latitude, 0 degrees longitude. After a quarter of a year, the Earth must rotate another quarter of a rotation (90 degrees) for the Sun to be overhead at that spot. Without that extra quarter of a rotation, the Sun is overhead at 90 degrees east longitude. That makes sense; the Earth has moved 90 degrees in its orbit. With a perfectly circular orbit, the rate at which this point moves eastward per day is constant, a little under a degree per day.
These points define a great circle around the Earth that coincides with the equator. A great circle is a circle on the surface of a sphere whose center is also the center of the sphere; it's the biggest possible circle that can exist on the surface of a sphere.
A Large Axial Tilt
Let's keep Earth's perfectly circular orbit, but let's give it a very large axial tilt, 80 degrees, for the purpose of this explanation. This Earth now begins at noon on the northward equinox at 0 degrees latitude, 0 degrees longitude. After one Earth rotation, where on Earth is the Sun overhead now? It's still a little less than one degree of angle away, but the direction of the displacement has changed. Instead of this point moving due east, it is mostly northward and only a little bit east. The Earth needs to rotate far less than 4 more minutes for the Sun to reach local noon for our observer at 0 degrees latitude, 0 degrees longitude. These points are west of where they would be without axial tilt. Solar noon has occurred before 24 hours have passed. As days continue to pass after the equinox, solar noons continue to occur earlier each solar day, as these points continue mostly northward and only a little bit eastward. The analemma shape as seen in the Northern hemisphere sky is moving up and to the right (west is to the right when looking south towards the Sun in the Northern hemisphere).
A quarter of a year later, it's the northern solstice, so let's see what has happened to our sun-overhead plots as the sidereal days have piled up. These points have continued to move mostly northward and a little eastward, but now they are moving purely eastward, because at the solstice the sun has stopped moving northward in the sky. It's still moving at a little less than 1 degree per day along that great circle, but because longitude lines are spaced much more closely together at 80 degrees latitude, one degree along a great circle covers many degrees of longitude. In other words, the eastward movement of these points has been "catching up" with the longitude of the corresponding no-axial tilt points. At the solstice, both the no-axial tilt point and the axial tilt point have the same longitude, 90 degrees east. The analemma shape as seen in the Northern hemisphere sky is moving up and to the left (eastward), where it reaches its highest point.
These points will continue to move through longitude lines rapidly while the Sun's overhead point remains far in the north. Now, the sun's overhead point is further east than it would be with no axial tilt. The analemma shape as seen in the Northern hemisphere sky is moving down and to the left (eastward).
As the southward equinox approaches, the sun-overhead points are moving mostly southward and only a little eastward. The analemma shape as seen in the Northern hemisphere sky is moving down and to the right (westward). This allows the non-axial tilt points to "catch up", and the longitude differences begin to decrease again, until the time of the southward equinox comes, when the two points coincide again.
Here is a crude ASCII drawing of these points so far, with longitude along the horizontal direction, and latitude along the vertical direction.
(2)
K L M N O
J P
I Q
H R
G S
(1)F T(3)
E U
D V
C W
B X
A B C D E F G H I J K L M N O P Q R S T U V W X Y
A = northward equinox
M = northern solstice
Y = southward equinox
The points around (1) are further west than their corresponding points on the equator. At (2), the points have caught up in longitude. At (3), they are further east than their corresponding points on the equator, but the equator points are catching up.
After the southward equinox, the longitudes of the sun-overhead point diverge once again, with the same mechanism as described above for the northward equinox. Just reverse north and south, and the eastward movements are the same. The analemma shape as seen in the Northern hemisphere sky first moves down and to the right, then down and to the left where it reaches its lowest point at the southern solstice, then up and to the left, then up and to the right where it reaches its starting point at the northward equinox.
Back to Reality
With an axial tilt of 80 degrees, the equation of time would show some extreme values, approaching almost 6 hours of divergence from the no-axial tilt case. With the true axial tilt of about 23.5 degrees, our equation of time difference values are far less substantial, but the effect is real.
The true analemma shape we see on Earth is the combined effect of the axial tilt that we have seen plus the fact that Earth's orbit is elliptical and it slows down during the parts of its orbit when it's farther from the Sun and it speeds up during the parts of its orbit when it's closer to the Sun.
The site Analemma has good explanations about the individual effects of the elliptical orbit and the axial tilt and how they combine to create our analemma.
