Ascribing computational principles to neural feedback circuits is an important problem in theoretical neuroscience. We study symmetric threshold-linear networks and derive stability results that go beyond the insights that can be gained from Lyapunov theory or energy functions. By applying linear analysis to subnetworks composed of coactive neurons, we determine the stability of potential steady states. We find that stability depends on two types of eigen-modes. One type determines global stability and the other type determines whether or not multistability is possible. We can prove the equivalence of our stability criteria with criteria taken from quadratic programming. Also, we show that there are permitted sets of neurons that can be coactive at a steady state and forbidden sets that cannot. Permitted sets are clustered in the sense that subsets of permitted sets are permitted and supersets of forbidden sets are forbidden. By viewing permitted sets as memories stored in the synaptic connections, we can provide a formulation of long-term memory that is more general than the traditional perspective of fixed point attractor networks.