The monoclinic double perovskite Sr2YRuO6 has recently gained a renewed interest in order to get a deeper insight into the exotic magnetic ground states associated with geometric frustration. Striking discrepancies between the spin order derived from the neutron diffraction refinements and the macroscopic magnetic and thermal responses is a major challenge that must be addressed. In this work, detailed neutron diffraction measurements as a function of temperature yield a completely different interpretation of the patterns. We show that at low temperatures a spin structure of the K2NiF4-type is an accessible configuration for the magnetic ground state. In the neighborhood of the magnetic transition, this configuration evolves into a canted superstructure. The deduced temperature dependence of the canting angle exhibits two closely spaced peaks, which are in excellent agreement with the double peaks in the magnetic contribution to the specific heat and in the thermal expansion coefficient. We explain these features in terms of reorientation of the net ferromagnetic moment of the noncollinear spin state, due to the local breaking of the inversion symmetry promoted by the monoclinic distortions, with structural changes acting as the driving force.