  Starting with a choice of a gauge group in four dimensions, there is often
freedom in the choice of magnetic and dyonic line operators. Different
consistent choices of these operators correspond to distinct physical theories,
with the same correlation functions of local operators in R^4. In some cases
these choices are permuted by shifting the theta-angle by 2pi. In other cases
they are labeled by new discrete theta-like parameters. Using this
understanding we gain new insight into the dynamics of four-dimensional gauge
theories and their phases. The existence of these distinct theories clarifies a
number of issues in electric/magnetic dualities of supersymmetric gauge
theories, both for the conformal N=4 theories and for the low-energy dualities
of N=1 theories.
