What is the correct term to describe direction of travel on a curve that intersects itself exactly once? In a non-self-intersecting curve such as a circle, we can describe direction of travel as clockwise (CW) or counter-clockwise (CCW).
In a self-intersecting curve such as a figure-eight, clockwise is no longer sensible, because a traveler on the curve alternates between clockwise and counter-clockwise at each pass through the intersection.
Is there a standard mathematical term that describes the direction of travel on a figure-eight?
My hunch is that if there is a standard term, then it would need to also describe how the curve is positioned/rotated in space.  For example, traveling on the infinity symbol, a traveler might be traveling downward when crossing the intersection.  If the curve is rotated 180 degrees, the same traveler would then be traveling upward when crossing the intersection.
If there is no standard term, then what term can I use to meaningfully communicate with other people which direction a traveler is going on a figure-eight with fixed rotation in 2D space like the infinity symbol?
 A: Direction of travel by itself is meaningless. You have to fix a point to talk about direction of travel. If you define a term "zigward" and "zagward", and said, for example "it's going zigward" then depending on the point of view of the observer there are parts of the figure of eight where it could appear to be travelling in either direction.
Take the purple and blue arrows in the following image. For observer A they are going in the opposite direction to what observer B sees, so A's zigward is B's zagward. You have to fix a point or another and then use that to describe the direction of motion relative to it.

There is no way that you can communicate a "direction" around a figure of eight to someone without a reference point. "Clockwise" and "anticlockwise" work because the centre of the circle forms a natural reference point, and we don't really think about "clockwise relative to the centre of the circle" but you could move the reference point to outside the circle, and at some areas of the circle a "clockwise" rotation would be going in the opposite direction to "clockwise".
Japanese people use "goes right" for clockwise and "goes left" for anticlockwise, which makes just as much sense really.
A: Most probably, you would like the description to be invariant under a rotation and translation of the curve - because the direction of travel on a circle is invariant under rotation and translation of the circle. And more generally, invariant under an orientation-preserving homeomorphism of the plane.
These conditions seem to preclude any position/rotation indicator of the curve. To take an example, the infinity symbol is globally invariant when rotated by $\pi$, but the direction of travel on this symbol is inverted.
So there seems to be few solutions:

*

*Distinguish the two closed parts of the curve by pointing one of them, i.e. by choosing a point (which is not the intersection point, of course). The concept of a pointed space is already used in topology. A pointed infinity symbol needs to be rotated $2\pi$, not just $\pi$, to be globally invariant.

*Distinguish the place where the trajectory starts, also by pointing it (and it must of course be another point than the intersection point). In many cases one is looking at a path, i.e. which has a begin point and an end point, so this seems handy.

