Yes, it does. And that’s true no matter where you live on Earth, because we all see the same sky.
No matter where you are on Earth (except at the North and South Poles), you have a due east and due west point on your horizon. That point marks the intersection of your horizon with the celestial equator – the imaginary line above the true equator of the Earth. At the equinoxes, the sun appears overhead at noon as seen from Earth’s equator. That’s the definition of an equinox: it’s when the sun crosses the celestial equator, as seen in Earth’s sky.
That’s why the sun rises due east and sets due west for all of us on the day of an equinox. The sun is on the celestial equator, and the celestial equator intersects all of our horizons at points due east and due west.
This fact makes the day of an equinox a good day for finding due east and due west from your yard or other favorite site for watching the sky. Just go outside around sunset or sunrise and notice the location of the sun on the horizon with respect to familiar landmarks.
If you do this, you’ll be able to use those landmarks to find those cardinal directions in the weeks and months ahead, long after Earth has moved on in its orbit around the sun, carrying the sunrise and sunset points southward.