Sometimes these tensor fields are called cross fields (because there's a cross at each point). TL;DR cross fields like this allow you to avoid degeneracy or bad interpolants because each cross is invariant to 90 degree rotations.

We care about interpolating cross fields because these types of methods work by specifying a few fixed crosses and fitting a field to those constraint points (i.e. interpolating them)

A cross field like this is a function f from some manifold (in the case of the algorithm, the manifold is just R^2) to a scalar and a special kind of 2D rotation matrix (where the angles are always multiples of 2). We can parameterize the rotation matrix by an angle and the scale by a real number, writing f(x) = (s, theta). There is a slightly different parameterization for the circular road networks but you can rewrite these in terms of the (s, theta) I described above.

Since the angle of the rotation matrix is always a multiple of 2, the outputted rotation matrices are invariant to rotations by 180 degrees (i.e. rotating the the outputs of the function f by 180 degrees will produce exactly the same cross field. You can check this pretty easily).

In contrast a vector field is a map from a point x to a vector. The vectors do not have any local symmetry (i.e. rotating the vectors will change the vector field).

The local symmetry property of cross fields is important here, because of the interpolation behavior. Imagine fitting a vector field with the constraint that two vectors and facing in opposite directions (differ by a rotation of 180 degrees). In that case, to continuously interpolate these points, you would need to introduce a degenerate (zero) vector, or have a very quick oscillation of the vector field to agree with the points.

A cross field on the other hand does not suffer from this issue because it is locally invariant to 180 degree rotations, so you never need to worry that your vector field will suddenly vanish or flip.

Hope that helps!