We report observations of a striking reversal in the direction of electroosmotic flow (EOF) outside a conical glass nanopore as a function of salt concentration. At high ionic strengths we observe EOF in the expected direction as predicted by classical electrokinetic theory, while at low salt concentrations the direction of the flow is reversed. The critical crossover salt concentration depends on the pore diameter. Finite-element simulations indicate a competition between the EOF generated from the inner and outer walls of the pore, which drives flows in opposite directions. We have developed a simple analytical model which reveals that as salt concentration is reduced, the flow rates inside the pore are geometrically constrained, whereas there is no such limit for flows outside the pore. This model captures all the essential physics of the system and explains the observed data, highlighting the key role the external environment plays in determining the overall electroosmotic behaviour.