Why solar eclipse paths are symmetrical?

The orbit of the moon is inclined by 5.14° to the ecliptic.

As you may know, the ecliptic is the apparent path of the Sun across the sky, so it is also the plane of Earth's orbit around the Sun. The plane of the Moon's orbit is inclined by 5.14° to the plane of the Earth's orbit. The intersection of the two plans is a line bisecting both planes.

There is a nice picture of this at http://www.cnyo.org/tag/orbital-plane/. [I'll insert it here after checking the copyright]

So, it is only when the Earth is very near to the intersection line that an eclipse can occur, because it is only then that the Sun, Moon and Earth can be in a strait line. This occurs for slightly more than a month (called an eclipse season), twice a year.

Another angle comes into the picture too. The axis of the Earth is tilted, so the equator is at 23.4° to the plane of the ecliptic.

So, the pattern you have observed comes about because of that geometry, where mirror image eclipse paths occur about six months apart.

The linked pages explain a lot more of the complexities.