How Do I Know Whether Waveguides Support TE or TM Modes? I'm working through the formalism of waveguides, and I've seen many different situations with various boundary conditions: hollow metal rectangular waveguide, dielectric slab, dielectric rectangular waveguide, dielectric circular waveguide. Some of these support TE and TM modes and some don't. For the channel waveguides (rectangular and circular, not the slab ones), it seems like a good rule of thumb is this: you can get TE and TM modes if your waveguide is a conductor, but if your waveguide is a dielectric surrounded by another dielectric, you don't get TE and TM modes.
My question is: what is the intuition behind this last sentence?
Part of the difficulty is that many treatments of the situations where there are TE/TM modes simply say "let's set $E_z=0$ or $H_z=0$." And then they don't say that in the dielectric cases, and I don't know what stops me from doing that in the dielectric case.
Some math will undoubtedly be helpful, but I'm really looking for the intuition as well.
That's my basic question: how can I know intuitively when a waveguide supports TE/TM modes, or what stops me from simply setting (one of) the longitudinal field components to zero in a dielectric waveguide?
Edited for clarity.
 A: I cannot be sure the following is relevant, and you don't consider some specific waveguide, so let me discuss a situation I am familiar with: diffraction of a (cylindrical) electromagnetic wave on a circular cylinder. This problem was solved exactly by Wait.
It turns out that for a perfect conductor an incident TM (TE) wave creates a refracted TM (TE) wave, however this is not true for a dielectric and oblique incidence, so, for example, an incident TM wave creates both refracted TM and TE waves. So it is possible that in some cases there just cannot be pure TM or TE guided waves in a waveguide, depending on the electric properties of the latter.
EDIT (Sept 16, 2022) : Some confirmation of the above from Journal of the Optical Society of America Vol. 51, Issue 5, pp. 491-498 (1961):

In a metallic guide there are two sets of solutions, the transverse
electric and transverse magnetic modes. In the dielectric guide all
but the cylindrically symmetric modes TE$_{\mathrm{om}}$ and TM$_{\mathrm{om}}$ are
hybrid; i.e., they have both electric and magnetic $z$ components.

I suspect something similar takes place for rectangular waveguides.
A: In a longitudinally homogeneous waveguide if it is filled uniformly with linear dielectric or paramagnetic material you will get TE and TM modes, as you can find it in any book on waveguides. If you maintain the longitudinal homogeneity but transversally the permittivity and/or permeability may vary then the propagating wave must have a longitudinal component. This you can see if you assume piecewise constant filling and try to match the fields tangential/normal to the jump. More formally you can derive it by using the so-called Marcuvitz-Felsen equations, see Section 2.2a in Felsen-Marcuvitz. What is interesting about this is that you obtain a pair of vector differential equations (transmission line equations) for the 4 transversal components  without the longitudinals showing up the longitudinal components being proportional to the to the transversal curls. That you can reduce the original six-variable Maxwell's equation to four unknowns can be expected since we know that the ME can equally be represented by a scalar (1) and a vector (3) potential, altogether 1+3 = 4 unknowns.
Interestingly, and rather unexpectedly, at the discrete frequencies at which the propagating wave stops propagating, i.e, at cut-off, but not below where the mode is evanescent, the longitudinal field components may become zero. You can prove that to yourself, not too difficult, using the same Marcuvitz-Felsen equations, and check the components at cut-off. (If you need help how to do this with the MF equations then ask a separate question and I will try to respond to it.)
