  It is well known that a spinning body moving in a fluid suffers a force
orthogonal to its velocity and rotation axis --- it is called the Magnus
effect. Recent simulations of spinning black holes and (indirect) theoretical
predictions, suggest that a somewhat analogous effect may occur for purely
gravitational phenomena. The magnitude and precise direction of this
"gravitational Magnus effect" is still the subject of debate. Starting from the
rigorous equations of motion for spinning bodies in General Relativity
(Mathisson-Papapetrou equations), we show that indeed such an effect takes
place and is a fundamental part of the spin-curvature force. The effect arises
whenever there is a current of mass/energy, non-parallel to a body's spin. We
compute the effect explicitly for some astrophysical systems of interest: a
galactic dark matter halo, a black hole accretion disk, and the
Friedmann-Lema\^itre-Robertson-Walker (FLRW) spacetime. It is seen to lead to
secular orbital precessions potentially observable by future astrometric
experiments and gravitational-wave detectors. Finally, we consider also the
reciprocal problem: the "force" exerted by the body on the surrounding matter,
and show that (from this perspective) the effect is due to the body's
gravitomagnetic field. We compute it rigorously, showing the matching with its
reciprocal, and clarifying common misconceptions in the literature regarding
the action-reaction law in post-Newtonian gravity.
