  In this paper we study the local magnetic ray transform of symmetric tensor
fields up to rank two on a Riemannian manifold of dimension $\geq 3$ with
boundary. In particular, we consider the magnetic ray transform of the
combinations of tensors of different orders due to the nature of magnetic
flows. We show that such magnetic ray transforms can be stably inverted, up to
natural obstructions, near a strictly convex (with respect to magnetic
geodesics) boundary point. Moreover, a global invertibility result follows on a
compact Riemannian manifold with strictly convex boundary assuming that some
global foliation condition is satisfied.
