Buoyed by advances in space technology, several firms are planning satellite constellations to offer broadband Internet service. While these developments are happening quickly, there are also many uncertainties about the design of these networks. A key open question is whether or not they will incorporate direct connectivity between satellites, instead of only ground-satellite connections. We compare the network behavior resulting from the two outcomes of that question. Our analysis shows that inter-satellite links substantially reduce the temporal variations in latency, add greater resilience to weather, and could yield more than 3× the throughput achieved without such links. Thus, whether this one design element pans out could have a large bearing on the performance, reliability, and economics of these networks. CCS CONCEPTS • Networks → Physical links; Network performance modeling.