The advent of gravitational-wave astronomy motivates detailed study of the classical gravitational dynamics of spinning black holes in binary systems. Analytical calculations to those ends (for arbitrary mass ratios) have traditionally focused on post-Newtonian perturbation theory (expansion in 1/c), naturally paired with an expansion in small spins. I will discuss a complimentary approach using post-Minkowskian perturbation theory (expansion in G), paired with a nonperturbative treatment of the spins. New results for two-spinning-black-hole scattering at first post-Minkowskian order (linear order in G), given in simple closed forms to all orders in spin, reveal remarkable relationships between arbitrary-mass-ratio two-body dynamics and test-body dynamics in a fixed background, and between the dynamics of spinning and nonspinning black holes.