  We make use of the conformal compactification of Minkowski spacetime $M^{\#}$
to explore a way of describing general, nonlinear Maxwell fields with conformal
symmetry. We distinguish the inverse Minkowski spacetime $[M^{\#}]^{-1}$
obtained via conformal inversion, so as to discuss a doubled compactified
spacetime on which Maxwell fields may be defined. Identifying $M^{\#}$ with the
projective light cone in $(4+2)$-dimensional spacetime, we write two
independent conformal-invariant functionals of the $6$-dimensional Maxwellian
field strength tensors -- one bilinear, the other trilinear in the field
strengths -- which are to enter general nonlinear constitutive equations. We
also make some remarks regarding the dimensional reduction procedure as we
consider its generalization from linear to general nonlinear theories.
