The dynamics due to a periodic forcing (harmonic axial oscillations)
in a Taylor-Couette apparatus of finite length is examined numerically
in an axisymmetric subspace. The forcing delays the onset of
centrifugal instability and introduces a $Z_2$ symmetry that involves
both space and time. This paper examines the influence of this
symmetry on the subsequent bifurcations and route to chaos in a
one-dimensional path through parameter space as the centrifugal
instability is enhanced. We have observed a well-known route to chaos
via frequency locking and torus break-up on a 2-tori branch once the
$Z_2$ symmetry has been broken. However, this branch is not connected
in a simple manner to the $Z_2$-invariant primary branch. An
intermediate branch of 3-tori solutions, exhibiting heteroclinic and
homoclinic bifurcations, provides the connection. On this 3-tori
branch, a new gluing bifurcation of 3-tori is seen to play a central
role in the symmetry breaking process.