Food rheology is key to manage the swallowing safety of people affected by swallowing disorders (dys- phagia). Simple approaches to assess the flow properties of texture-modified drinks are widely used, but relatively poorly understood. This study focuses on the International Dysphagia Diet Standardisation Initiative (IDDSI) flow test, adopted by caregivers worldwide. This test considers the gravity-driven flow in a vertical syringe. Newtonian liquids and non-Newtonian fluids obtained using a commercial starch-based thickener were considered in this study. An accurate theoretical description of the flow is proposed for Newtonian and power-law fluids considering the effect of fluid properties and of the syringe geometry. A rheological map is proposed, based on the results of several thousand simulations, to capture quantitatively the effect of rheological properties and density on the IDDSI classification, highlighting the important effect of the fluid density which is usually ignored. The sensitivity of the IDDSI results with respect to the syringe outlet diameter is discussed, as well as the different average shear rates at which different IDDSI Levels are tested. The rheological map also shows quantitatively that different combinations of the fluid rheological properties and density can result in the same IDDSI classification, suggesting interesting directions for future clinical research.