When does a line drawing have an open unicursal tracing ?

An open unicursal tracing of a line drawing is equivalent to an Euler Path of a connected graph (with different starting and ending points).
Euler's Theorems tell us that a connected graph has an Euler Path (with different starting and ending points) if and only if there are exactly two vertices of odd degree.
So, if we think of each intersection of edges in a line drawing as a vertex, we can apply Euler's Theorems to tell us the following:

A line drawing has an open unicursal tracing if and only if it has exactly two points of intersection of odd degree.