But there's only one field there, the electromagnetic field. It's three-dimensional, and isotropic around an electron, and to depict a slice through it we have to combine the radial and circular field lines like this:
[qimg]http://www.jbum.com/pixmagic/pinwheel.jpg[/qimg]
OK, I understand this now and it's worth jumping back into the thread. Those spiral lines you've drawn are meant to be the magnetic field vectors PLUS the electric field vectors. The electric part goes out, the magnetic part goes around, so you think the electromagnetic whole thing goes in a spiral. Presumably if you had an E field pointing north and a B field pointing west, you'd think that the "electromagnetic field" pointed northwest, right?
This is nonsense. The full electromagnetic field (E and B together) is not a vector, it's a tensor. It is not the simple vector sum of E and B. In a combined E and B field, as from a moving electron, there is no quantity whatsoever that has a spiral vector direction.