When does a line drawing have a closed unicursal tracing ?

A closed unicursal tracing of a line drawing is equivalent to an Euler Circuit of a connected graph.
Euler's Theorems tell us that a connected graph has an Euler Circuit if and only if there are no 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 a closed unicursal tracing if and only if it has no points of intersection of odd degree.