  The rigid rotor is a classic problem in quantum mechanics, describing the
dynamics of a rigid body with its centre of mass held fixed. The configuration
space of this problem is $SO(3)$, the space of all rotations in three
dimensions. This is a topological space with two types of closed loops: trivial
loops that can be adiabatically shrunk to a point and non-trivial loops that
cannot. In the traditional formulation of the problem, stationary states are
periodic over both types of closed loops. However, periodicity conditions may
change if Berry phases are introduced. We argue that time-reversal-symmetry
allows for only one new possibility -- a Berry phase of $\pi$ attached to all
non-trivial loops. We derive the corresponding stationary states by exploiting
the connection between $SO(3)$ and $SU(2)$ spaces. The solutions are
anti-periodic over any non-trivial loop, i.e., stationary states reverse sign
under a $2\pi$ rotation about any axis. Remarkably, this framework is realized
in the low-energy physics of certain quantum magnets. The magnets must satisfy
the following conditions: (a) the classical ground states are unpolarized,
carrying no net magnetization, (b) the set of classical ground states is
indexed by $SO(3)$, and (c) the product $N\times S$ is a half-integer, where
$N$ is the number of spins and $S$ is the spin quantum number. We demonstrate
this result in a family of Heisenberg antiferromagnets defined on polygons with
an odd number of vertices. At each vertex, we have a spin-$S$ moment that is
coupled to its nearest neighbours. In the classical limit, these magnets have
coplanar ground states. Their quantum spectra, at low energies, correspond to
`spherical top' and `symmetric top' rigid rotors. For integer values of $S$, we
recover traditional rigid rotor spectra. With half-integer-$S$, we obtain rotor
spectra with a Berry phase of $\pi$.
