The short answer is no, there are no cases where radio waves would not have orthogonal magnetic and electric fields.
In physics, a radio wave, indeed all EM radiation is called a transverse wave, meaning, by definition, that the oscillations of the waves are perpendicular to the direction of energy transfer and travel.
The electric and magnetic parts of the field stand in a fixed ratio of strengths in order to satisfy the two Maxwell equations that specify how one is produced from the other. These $\mathbf{E}$ and $\mathbf{B}$ (in physics the magnetic part uses B for some reason, I'll maintain that convention to make it easier for the physics folks to correct any errors) fields are also in phase, with both reaching maxima and minima at the same points in space.
OK, so we know this much already, but why?
Ask James Clerk Maxwell; his electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field E or the magnetic field B, takes the form:
$$
\left(\nabla^2 - { \mu\epsilon } {\partial^2 \over \partial t^2} \right) \mathbf{E} = \mathbf{0}
\\
\left(\nabla^2 - { \mu\epsilon } {\partial^2 \over \partial t^2} \right) \mathbf{B} = \mathbf{0}
$$
where
$$c = {1 \over \sqrt {\mu\epsilon} }$$
is the speed of light in a medium with permeability ($\mu$), and permittivity ($\epsilon$), and ∇2 is the Laplace operator. In a vacuum, c = 299,792,458 meters per second, which is the speed of light in free space.
So that's the general theory, this can be solved for the plane wave case (the one we are interested in):
Given a plane defined by a unit normal vector: $\mathbf{n} = { \mathbf{k} \over k }$.
Then planar traveling wave solutions of the wave equations are:
$$\mathbf{E}(\mathbf{r}) = E_0 e^{ -i \mathbf{k} \cdot \mathbf{r} }$$
and
$$\mathbf{B}(\mathbf{r}) = B_0 e^{ -i \mathbf{k} \cdot \mathbf{r} }$$
where r = (x, y, z) is the position vector (in meters).
These solutions represent planar waves traveling in the direction of the normal vector n. If we define the z direction as the direction of n. and the x direction as the direction of E., then by Faraday's Law the magnetic field lies in the y direction and is related to the electric field by the relation $\scriptstyle c^2{\partial B \over \partial z} \,=\, {\partial E \over \partial t}$. Because the divergence of the electric and magnetic fields are zero, there are no fields in the direction of propagation.
References