  It has recently been emphasized that all known exact evaluations of gap
probabilities for classical unitary matrix ensembles are in fact
$\tau$-functions for certain Painlev\'e systems. We show that all exact
evaluations of gap probabilities for classical orthogonal matrix ensembles,
either known or derivable from the existing literature, are likewise
$\tau$-functions for certain Painlev\'e systems. In the case of symplectic
matrix ensembles all exact evaluations, either known or derivable from the
existing literature, are identified as the mean of two $\tau$-functions, both
of which correspond to Hamiltonians satisfying the same differential equation,
differing only in the boundary condition. Furthermore the product of these two
$\tau$-functions gives the gap probability in the corresponding unitary
symmetry case, while one of those $\tau$-functions is the gap probability in
the corresponding orthogonal symmetry case.
