This article deals with strong structural controllability of leader-follower networks. The system matrix defining the network dynamics is a pattern matrix, in which a priori given entries are equal to zero, while the remaining entries take nonzero values. These nonzero entries correspond to edges in the network graph. The network is called strongly structurally controllable if for all choices of real values for the nonzero entries in the pattern matrix, the system is controllable in the classical sense. The novelty of this article is that we consider the situation that prespecified nonzero entries in the system's pattern matrix are constrained to take identical (nonzero) values. These constraints can be caused by symmetry properties or physical constraints on the network. Restricting the system matrices to those satisfying these constraints yields a new notion of strong structural controllability. The aim of this article is to establish graph-theoretic conditions for this more general property of strong structural controllability.