Classical electromagnetism is perfectly parity-invariant; it is not intrinsically left-handed or right-handed. It is true that you need to use the right hand rule to find the magnetic field, but you need to use it again to find the magnetic force. Since you always use it twice to get any directly observable quantity, the minus sign you would pick up from using the left hand rule cancels.
If we phrase everything in terms of forces, then electrostatics and magnetostatics reduce to the facts that (1) like charges repel and (2) parallel currents attract. This is clearly independent of any handedness convention. (Incidentally, the relative sign here comes from the relative sign between time and space in relativity.)
There's some deeper mathematics lurking under the surface here. Recall that the cross product of two vectors is defined as the vector pointing perpendicular to the parallelogram formed by the two vectors, with the same length as the area of the parallelogram. In three dimensions, there are two directions perpendicular to every parallelogram, which is why we need the right hand rule to pick one. In higher dimensions, this definition doesn't work at all, because there are infinitely many directions perpendicular to every plane.
Hence the magnetic field in general dimensions can't be thought of as a vector. Instead it's better to just say it is the parallelogram itself -- it is a plane and area at every point, rather than a direction and length like a vector. The magnetic force just causes particles to rotate in the plane of the field. A current $\mathbf{J}$ at the origin creates a magnetic field at $\mathbf{r}$ in the plane spanned by $\mathbf{r}$ and $\mathbf{J}$.
Formally, these area elements are called rank $2$ differential forms. They're too involved for a beginning course, which is why we instead use the right-hand rule to convert the area to a vector, introducing an arbitrary choice. But all of the physics can be written in a manifestly symmetric way, because the phenomena really are symmetric.
