  Symmetry is a fundamentally important concept in many branches of physics. In
this work, we discuss two types of symmetries, external symmetry and internal
symmetry, which appear frequently in controlled quantum spin chains and apply
them to study various controllability problems. For spin chains under single
local end control when external symmetries exists, we can rigorously prove that
the system is controllable in each of the invariant subspaces for both XXZ and
XYZ chains, but not for XX or Ising chains. Such results have direct
applications in controlling antiferromagnetic Heisenberg chains when the
dynamics is naturally confined in the largest excitation subspace. We also
address the theoretically important question of minimal control resources to
achieve full controllability over the entire spin chain space. In the process
we establish a systematic way of evaluating the dynamical Lie algebras and
using known symmetries to help identify the dynamical Lie algebra.
