Suppose you wanted to draw pictures of how the stars move. You go out on a clear night with a toilet paper tube (because you want to mask out distractions, and you don't trust lenses), and you look through it (for hours at a time, long enough for the stars to move a long way in the sky) in different directions. As it happens, you're in the northern hemisphere at a moderate latitude, like in central Illinois. (If you were elsewhere, such as Australia, the individual pictures would look different, but the overall phenomena described would work the same.)
If you look northwest, you'll see the stars moving like this:
Why? Because you're looking at the lower left quadrant of this:
Which is the northern stars circling counterclockwise around a hub near Polaris. which is visible to you in the northern sky. (That is, if you point your toilet paper tube due north, and up at about a 40 degree angle, you'll see the North Star and other stars making counterclocwise circles around it, like near the top right of the above photo). That hub is the projection of the earth's axis of rotation onto the apparent celestial sphere.
If you look southwest, you'll see the stars moving like this instead:
This is because, like in the northwest view, you're seeing stars rotating around a hub, in this case the celestial south pole. It looks different, though, for two reasons. One is that in this case the hub is well below the horizon, so you're seeing the upper right quadrant of the rotation instead of the lower left quadrant. The other is that the rotation is clockwise instead of counterclockwise. (The celestial sphere is one big rotating thing that all rotates in the same direction, but you're now looking toward the other end of it. Imagine standing inside a giant barrel placed horizontally and tumbling, that is, rotating around its axis. These used to be commmon in funhouses. Suppose the barrel is rotating counterclockwise as you face toward the end of the barrel that's ahead of you. If you turn around and look behind you, it will be rotating counterclockwise. Clockwise and counterclockwise are always relative to your facing. Even the hands of a conventional clock would go counterclockwise, if the workings of the clock were transparent and you looked at the hands from behind the clock face.)
Finally, you look due west. You'll see the stars moving like this:
These stars are near the celestial equator. They're making circles too, just like the northern stars and the southern stars, but for these circles, you're standing at the center of the circles. You are standing on (or so near as to make literally no difference at all, a mere few thousand miles compared to the effectively infinite distances of the stars) the axis they're rotating around. So their paths appear to be straight lines, just like a disc appears to be a straight line when you look at it edge-on, and just like the horizon appears to be a straight line even though it is encircling you.
However, if you could measure more closely (you'd have to mount your toilet paper tube on a sturdy tripod, instead of free-handing it resulting in the wiggly and not very accurate lines I drew), only stars that are exactly on the celestial equator make exactly straight lines. For the ones a little south (left) of the equator, you're standing a little to the right of the centers of their circles, so they'll appear to curve away from you just a little to the left as they set. And similarly for the ones a little north (right) of the equator; they'll appear to curve away from you just a little to the right as they set.
Now, having studied the motions of the stars in three different directions, you decide you want to plaint a big panorama painting that shows all that motion on one wide canvas. You don't want to distort anything; you want the painting to show what you actually see, so star trails way to the left of the painting (southwest) will curve more and more to the left as they get nearer the horizon, and star trails way to the right will curve more and more to the right as they get nearer the horizon, and star trails near the middle (west) will be nearly straight. You want a realistic painting (no cubism), so you realize that you need to show this as a continuous change across the width of the canvas, from left-curving to straight to right-curving.
As a result, the star trails you paint will not be parallel to one another. The trails will squeeze together near the center of the painting (especially, along a line from the lower left to the upper right corners), and seem to spread apart at the lower right and upper left corners, in the same way (and for the same reason) that many world map projections seem to squeeze the continents together near the equator and spread them apart nearer the poles.
If instead of a very wide panorama, you want only a typical wide angle camera view generally toward the west, the squeezing and spreading will be less dramatic, but you'll still see, going left to right across the frame, star trails curving to the left, then curving less, then straight, then curving right, then curving right more.
The effect will be something like this:
Note that this picture is looking east, where everything is rising instead of setting, and therefore is also tilted the opposite way, but it works the same way. This link to the image source has an explanation of what it shows and how it was made
. Quoting the page, the image "shows very well how the stars near the celestial Equator trace lines that are almost straight, while the stars at the North and South of the Equator, respectively, appear to draw circles between the celestial North and South poles."
Now, I turn your attention to edfel's excellent "time lapse" composite image of the OTC:
It shows all the things I've been talking about. The star trails curve to the right (as they get closer to setting) toward the right (more northerly) side of the frame. The farther right, the more they curve, but where they cross the tree they're curving very little, and the (rather faint) star trail that crosses near the base of the trunk of the tree looks very straight. The trails farthest left (also rather faint) curve to the left instead, though you might have to use a straightedge or a line drawing program to see it.
This makes me confident (though not absolutely certain) that the celestial equator crosses near the base of the tree trunk, and where that meets the horizon (which might not be exactly where it meets the visible ground) is true west.
There is another possibility: that the view is distorted; that is to say, distorted in a different way than what we usually expect. It would be possible, for instance, to find a projection (or physically craft a lens) that creates a view where one of the curved lines toward the right becomes straight. Then, we'd be fooled into thinking that's the celestial equator instead. That's why we sometimes qualify our conclusions with "...unless it's a projection error."