When do non-TEM waves occur?
If you check out the wikipedia page:

Transverse Mode

You'll see at the start of the second paragraph they say 

"Transverse modes occur because of boundary conditions imposed on the wave by the waveguide"

This accounts for the discrepancy you have observed. Most diagrams of electromagnetic waves depict waves in free space, far away from any objects, and in that situation, you will have no electric or magnetic component in the propagation direction. Waveguides are (metal or fiber) tubes that confine an electromagnetic wave and force it in a particular direction. Since you have now introduced materials into the picture, things get more complicated, and you can have a non-zero electric and magnetic fields in the propagation direction.

Most E+M textbooks will have a section on waveguides, and these concepts figure heavily into fiber optics engineering.