  In this article, we introduce and study various V-line transforms (VLTs)
defined on symmetric 2-tensor fields in $\mathbb{R}^2$. The operators of
interest include the longitudinal, transverse, and mixed VLTs, their integral
moments, and the star transform. With the exception of the star transform, all
these operators are natural generalizations to the broken-ray trajectories of
the corresponding well studied concepts defined for straight-line paths of
integration. We characterize the kernels of the VLTs and derive exact formulas
for reconstruction of tensor fields from various combinations of these
transforms. The star transform on tensor fields is an extension of the
corresponding concepts that have been previously studied on vector fields and
scalar fields (functions). We describe all injective configurations of the star
transform on symmetric 2-tensor fields and derive an exact, closed-form
inversion formula for that operator.
