Understanding the patterns of human mobility between cities has various applications from transport engineering to spatial modeling of the spreading of contagious diseases. We adopt a city-centric, data-driven perspective to quantify such patterns and introduce the mobility signature as a tool for understanding how a city (or a region) is embedded in the wider mobility network. We demonstrate the potential of the mobility signature approach through two applications that build on mobile-phone-based data from Finland. First, we use mobility signatures to show that the well-known radiation model is more accurate for mobility flows associated with larger Finnish cities, while the traditional gravity model appears a better fit for less populated areas. Second, we illustrate how the SARS-CoV-2 pandemic disrupted the mobility patterns in Finland in the spring of 2020. These two cases demonstrate the ability of the mobility signatures to quickly capture features of mobility flows that are harder to extract using more traditional methods.