  We study the scaling behaviour of a class of compatible two-well problems for
higher order, homogeneous linear differential operators. To this end, we first
deduce general lower scaling bounds which are determined by the vanishing order
of the symbol of the operator on the unit sphere in direction of the associated
element in the wave cone. We complement the lower bound estimates by a detailed
analysis of the two-well problem for generalized (tensor-valued) symmetrized
derivatives with the help of the (tensor-valued) Saint-Venant compatibility
conditions. In two spatial dimensions for highly symmetric boundary data (but
arbitrary tensor order $m \in \mathbb{N}$) we provide upper bound constructions
matching the lower bound estimates. This illustrates that for the two-well
problem for higher order operators new scaling laws emerge which are determined
by the Fourier symbol in the direction of the wave cone. The scaling for the
symmetrized gradient from \cite{CC15} which was also discussed in \cite{RRT23}
provides an example of this family of new scaling laws.
