  The well-known curvature method initiated in works of Melnikov and Verdera is
now commonly used to relate the $L^2(\mu)$-boundedness of certain singular
integral operators to the geometric properties of the support of measure $\mu$,
e.g. rectifiability. It can be applied however only if Menger curvature-like
permutations, directly associated with the kernel of the operator, are
non-negative. We give an example of an operator in the plane whose
corresponding permutations change sign but the $L^2(\mu)$-boundedness of the
operator still implies that the support of $\mu$ is rectifiable. To the best of
our knowledge, it is the first example of this type. We also obtain several
related results with Ahlfors-David regularity conditions.
