  We study singular integral operators with kernels that are more singular than
standard Calder\'on-Zygmund kernels, but less singular than bi-parameter
product Calder\'on-Zygmund kernels. These kernels arise as restrictions to two
dimensions of certain three-dimensional kernels adapted to so-called Zygmund
dilations, which is part of our motivation for studying these objects. We make
the case that such kernels can, in many ways, be seen as part of the extended
realm of standard kernels by proving that they satisfy both a T1 theorem and
commutator estimates in a form reminiscent of the corresponding results for
standard Calder\'on-Zygmund kernels. However, we show that one-parameter
weighted estimates, in general, fail.
