Longitudinal-section modes, additionally, have a component of either magnetic or electric field that is zero in one transverse direction.
When determining whether a structure can support a particular TE mode, one sets the electric field in the z direction (the longitudinal direction of the line) to zero and then solves Maxwell's equations for the boundary conditions set by the physical structure of the line.
One can just as easily set the electric field in the x direction to zero and ask what modes that gives rise to.
Thus, in general, the LSE and LSM modes have a longitudinal component of both electric and magnetic field.
Likewise the LSM modes are found by setting one of the transverse components of magnetic field to zero with analogous results.