  The basic ingredients of the `consistent histories' approach to quantum
theory are a space $\UP$ of `history propositions' and a space $\D$ of
`decoherence functionals'. In this article we consider such history quantum
theories in the case where $\UP$ is given by the set of projectors $\P(\V)$ on
some Hilbert space $\V$. We define the notion of a `physical symmetry of a
history quantum theory' (PSHQT) and specify such objects exhaustively with the
aid of an analogue of Wigner's theorem. In order to prove this theorem we
investigate the structure of $\D$, define the notion of an `elementary
decoherence functional' and show that each decoherence functional can be
expanded as a certain combination of these functionals. We call two history
quantum theories that are related by a PSHQT `physically equivalent' and show
explicitly, in the case of history quantum mechanics, how this notion is
compatible with one that has appeared previously.
