Do Maxwell's Equations depend upon an orientation of space?

It doesn't look like the equations are preserved under orientation-reversal.

Short answer: Maxwell’s equations are preserved under a parity transformation, but you have to be careful to transform quantities correctly.

We have to address two questions.

  1. What happens to the quantities involved in Maxwell’s equations upon a parity transformation?

The normal vector quantities: positions, velocities, forces, and $J$ (vector current density) transform in the usual way, they all flip sign. The electric field is also defined to change sign, but the magnetic field is an odd duck, it is defined to keep its sign (and is called a "pseudovector" for that reason).

In the covariant formulation, you can deduce how both fields (and $F$) transform from just one rule: the four-potential $A$ is defined to transform as a normal four-vector (i.e. the time component stays the same, the spacial components change sign).

Finally, in the language of differential forms, $F$ is a two-form, which is invariant under the parity transformation, and $J$ is a twisted three-form, which changes sign under parity.

  1. Do Maxwell’s equations still hold if we perform a parity transformation?

Yes, if we follow the rules in the previous paragraph the equations still hold. In fact that's why we define the magnetic field (or $A$ in the covariant formulation) to transform in this way.

In the language of differential forms, $\mathrm{d}\star F = J$ stays true under parity, because both sides change sign.

At the freshman level, we can visualize why we define the magnetic field to not change sign with a simple example. Imagine a long wire with a current going through it, and the magnetic field curling around it according to the right hand rule. What happens if you reflect everything about the origin? The current will of course change direction, but now you have to decide what you will do with the magnetic field. If you decide to transform it the way normal vectors transform, by changing its sign, then the resulting picture will become unphysical, the magnetic field will stop following the right hand rule. But if you follow the "odd duck" prescription for the magnetic field, then after the transformation everything will still look great.