  The Maxwell invariant plays a fundamental role in the mathematical
description of electromagnetic fields in charged spacetimes. We present a
detailed {\it analytical} study of the physical and mathematical properties of
the Maxwell electromagnetic invariant ${\cal F}_{\text{KN}}(r,\theta;M,a,Q)$
which characterizes the Kerr-Newman black-hole spacetime. It is proved that,
for all Kerr-Newman black-hole spacetimes, the spin and charge dependent
minimum value of the Maxwell electromagnetic invariant is attained on the
equator of the black-hole surface. Interestingly, we reveal the physically
important fact that Kerr-Newman spacetimes are characterized by two critical
values of the dimensionless rotation parameter ${\hat a}\equiv a/r_+$, ${\hat
a}^{-}_{\text{crit}}=\sqrt{3-2\sqrt{2}}$ and ${\hat a}^{+}_{\text{crit}}=
\sqrt{5-2\sqrt{5}}$, which mark the boundaries between three qualitatively
different spatial functional behaviors of the Maxwell electromagnetic
invariant: (i) Kerr-Newman black holes in the slow-rotation ${\hat a}<{\hat
a}^{-}_{\text{crit}}$ regime are characterized by negative definite Maxwell
electromagnetic invariants that increase monotonically towards spatial
infinity, (ii) for black holes in the intermediate spin regime ${\hat
a}^{-}_{\text{crit}}\leq {\hat a}\leq{\hat a}^{+}_{\text{crit}}$, the positive
global maximum of the Kerr-Newman Maxwell electromagnetic invariant is located
at the black-hole poles, and (iii) Kerr-Newman black holes in the
super-critical regime ${\hat a}>{\hat a}^{+}_{\text{crit}}$ are characterized
by a non-monotonic spatial behavior of the Maxwell electromagnetic invariant
along the black-hole horizon with a spin and charge dependent global maximum
whose polar angular location is characterized by the dimensionless functional
relation ${\hat a}^2\cdot(\cos^2\theta)_{\text{max}}=5-2\sqrt{5}$.
