  Let $E_1,\dots ,E_k$ and $E$ be natural vector bundles defined over the
category $\Cal Mf_m^+$ of smooth oriented $m$--dimensional manifolds and
orientation preserving local diffeomorphisms, with $m\geq 2$. Let $M$ be an
object of $\Cal Mf_m^+$ which is connected. We give a complete classification
of all separately continuous $k$--linear operators $D\:\Ga
_c(E_1M)\x\dots\x\Ga_c(E_kM)\to \Ga (EM)$ defined on sections with compact
supports, which commute with Lie derivatives, i\.e\. which satisfy $$ \Cal
L_X(D(s_1,\dots ,s_k))=\sum _{i=1}^kD(s_1,\dots ,\Cal L_Xs_i,\dots,s_k), $$ for
all vector fields $X$ on $M$ and sections $s_j\in\Ga_c(E_jM)$, in terms of
local natural operators and absolutely invariant sections. In special cases we
do not need the continuity assumption. We also present several applications in
concrete geometrical situations, in particular we give a completely algebraic
characterization of some well known Lie brackets.
