We consider the relation of symmetries and subspace controllability for spin-1/2 networks with XXZ couplings, subject to perturbation of a single node by a local potential (Z-control). The Hamiltonians for such networks decompose into excitation subspaces. Focusing on the single excitation subspace, it is shown for single-node Z-controls that external symmetries are characterized by eigenstates of the system Hamiltonian which have zero overlap with the control node, and there are no internal symmetries. It is further shown that there are symmetries which persist even in the presence of random perturbations. For XXZ chains with uniform coupling strengths, a characterization of all possible symmetries is given which shows a strong dependence on the position of the node we control. We then show for Heisenberg and XX chains with uniform coupling strength subject to single-node Z-control that the lack of symmetry is both necessary and sufficient for subspace controllability. Finally, the latter approach is generalized to establish controllability results for simple branched networks.