If there is an inertial frame where $\bf{E}=0$ (or $\bf{B}=0$), than in all other inertial frames will be either $\bf{E}=0$ (or $\bf{B}=0$), or $\bf{E} \perp \bf{B}$.
If, for instance, $\bf{B}=0$, than in this frame the charge is accelerated in the direction of $\bf{E}$. But from the other frame this dynamics is seen as combination of acceleration and rotation. This is because in Minkowski spase all that happens to vectors is either acceleration (change of energy and magnitude of spatial momentum) or rotation (change in the direction of spatial momentum).
Any acceleration (Lorentz boost) is interpreted as due to (transformed) $\bf{E}$, while the (3D) rotation is interpreted as due to (transformed) $\bf{B}$. That's why in the new frame we can see the field ($\bf{B}$ or $\bf{E}$) that was zero in the initial frame. But they will always be orthogonal due to transformation properties of the acceleration under Lorentz transformation, regardless of the properties of the sources of the fields $\bf{B}$ and $\bf{E}$.
On the pther hand, if $\bf{E} \perp \bf{B}$ and $E \neq B$, than there is an inertial frame where $\bf{E}=0$ or $\bf{B}=0$. The only case when fields are orthogonal in all frames is $\bf{E} \perp \bf{B}$ and $E = B$ (elecrtomagnetis waves in vacuum).
If the fields are not orthogonal in one frame, they will not be orthogonal in any other frame. In that case there is a reference frame where both $\bf{B}$ and $\bf{E}$ are parallel to each other, and acceleration (due to change in energy and magnitude of momentum) and centripetal (or centrifugal) force have the same direction in this frame.