Dimensionality reduction methods are widely used in neuroscience to investigate two complementary aspects of neural activity: the distribution of single-neuron functional properties and the low-dimensional collective dynamics of population activity. However, how do these two aspects of neural activity relate to the structure of the underlying neural circuit? In this work, we connect circuit structure, single-neuron functional properties, and emerging low-dimensional dynamics in spiking recurrent network models. Our models explain how topologically distinct circuit structures can produce equivalent low-dimensional dynamics. Despite this degeneracy, we find that circuit structure imposes specific constraints on both the low-dimensional dynamics of population activity and the distribution of single-neuron functional properties. These constraints yield simple criteria for comparing network models with observed neural activity. Our modeling framework not only links classical models of cortical circuits to the more recent notion of neural manifolds but also paves the way for designing tractable models of population dynamics that are better aligned with neural recordings.