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The Gauss-Bonnet curvature invariant has attracted the attention of physicists and mathematicians over the years. In particular, it has recently been proved that black holes can support external matter configurations that are non-minimally coupled to the Gauss-Bonnet invariant of the curved spacetime. Motivated by this physically interesting behavior of black holes in Einstein-Gauss-Bonnet theories, we present a detailed {\it analytical} study of the physical and mathematical properties of the Gauss-Bonnet curvature invariant ${\cal G}_{\text{Kerr}}(r,\cosθ;a/M)$ of spinning Kerr black holes in the spacetime region outside the horizon. Interestingly, we prove that, for all spinning Kerr spacetimes in the physically allowed regime $a/M\in[0,1]$, the spin-dependent maximum curvature of the Gauss-Bonnet invariant is attained at the equator of the black-hole surface. Intriguingly, we reveal that the location of the global minimum of the Gauss-Bonnet invariant has a highly non-trivial functional dependence on the black-hole rotation parameter: (i) For Kerr black holes in the dimensionless slow-rotation $a/M<(a/M)^{-}_{\text{crit}}=1/2$ regime, the Gauss-Bonnet curvature invariant attains its global minimum asymptotically at spatial infinity, (ii) for black holes in the intermediate spin regime $1/2=(a/M)^{-}_{\text{crit}}\leq a/M\leq(a/M)^{+}_{\text{crit}}= \sqrt{\Big\{{{7+\sqrt{7}\cos\Big[3^{-1}\arctan\big(3\sqrt{3}\big)\Big]- \sqrt{21}\sin\Big[3^{-1}\arctan\big(3\sqrt{3}\big)\Big]\Big\}}/12}}$, the global minima are located at the black-hole poles, and (iii) Kerr black holes in the super-critical regime $a/M>(a/M)^{+}_{\text{crit}}$ are characterized by a non-trivial functional behavior of the Gauss-Bonnet curvature invariant along the black-hole horizon with a spin-dependent polar angle for the global minimum point.