  We search for signs of rotation in the subsystems of the Milky Way and M31
that are defined by their satellite galaxies, their globular cluster
populations, and their BHB stars. A set of simple distribution functions are
introduced to describe anisotropic and rotating stellar populations embedded in
dark haloes of approximate Navarro-Frenk-White form. The BHB stars in the Milky
Way halo exhibit a dichotomy between a prograde rotating, comparatively
metal-rich component ([Fe/H] > -2) and a retrograde rotating, comparatively
metal-poor ([Fe/H] < -2) component. The prograde metal-rich population may be
associated with the accretion of a massive satellite (~ 10^9 solar masses). The
metal-poor population may characterise the primordial stellar halo and the net
retrograde rotation could then reflect an underestimate in our adopted local
standard of rest circular velocity Theta_0. If Theta_0 is ~ 240 km/s then the
metal-poor component has no rotation and there is a net prograde rotation
signal of ~45 km/s in the metal-rich component. There is reasonable evidence
that the Milky Way globular cluster and satellite galaxy systems are rotating
with <v_phi> ~50 km/s and <v_phi> ~ 40 km/s respectively. Furthermore, a
stronger signal is found for the satellite galaxies when the angular momentum
vector of the satellites is inclined with respect to the normal of the disc.
The dwarf spheroidal satellites of M31 exhibit prograde rotation relative to
the M31 disc with <v_phi> ~ 40 km/s. We postulate that this group of dwarf
spheroidals may share a common origin. We also find strong evidence for
systemic rotation in the globular clusters of M31 particularly for the most
metal-rich.
