The problem of finding all r-th order differential invariants of immersions of manifolds with metric fields, with values in a left (G^1_m x G^1_n)-manifold is formulated. For obtaining the basis of higher order differential invariants the orbit reduction method is used. As a new result it appears that r-th order differential invariants depending on an immersion f:M->N of manifolds M and N and on metric fields on them can be factorized through metrics, curvature tensors and their covariant derivatives up to the order (r-2), and covariant differentials of the tangent mapping Tf up to the order r. The concept of a covariant differential Tf is also introduced in this paper. The obtained results are geometrically interpreted as well.