  The Villain form of SO(3) lattice gauge theory is studied and compared to
Wilson's SU(2) theory. The topological invariants in SO(3) which correspond to
twisted boundary conditions in SU(2) are discussed and lattice observables are
introduced for them. An apparent SO(3) phase with negative adjoint Polyakov
loop is explained in terms of these observables. The electric twist free
energy, an order parameter for the confinement-deconfinement transition, is
measured in both theories to calibrate the temperature. The results indicate
that lattices with about 700^4 sites or larger will be needed to study the
SO(3) confined phase. Alternative actions are discussed and an analytic path
connecting SO(3) and SU(2) lattice gauge theory at weak coupling is exhibited.
The relevance for confinement of the centre of the gauge group is discussed.
