The highest superconducting transition temperatures in the cuprates are achieved in bilayer and trilayer systems, highlighting the importance of interlayer interactions for high T c. It has been argued that interlayer hybridization vanishes along the nodal directions by way of a specific pattern of orbital overlap. Recent quantum oscillation measurements in bilayer cuprates have provided evidence for a residual bilayer-splitting at the nodes that is sufficiently small to enable magnetic breakdown tunneling at the nodes. Here we show that several key features of the experimental data can be understood in terms of weak spin-orbit interactions naturally present in bilayer systems, whose primary effect is to cause the magnetic breakdown to be accompanied by a spin flip. These features can now be understood to include the equidistant set of three quantum oscillation frequencies, the asymmetry of the quantum oscillation amplitudes in c-axis transport compared to ab-plane transport, and the anomalous magnetic field angle dependence of the amplitude of the side frequencies suggestive of small effective g-factors. We suggest that spin-orbit interactions in bilayer systems can further affect the structure of the nodal quasiparticle spectrum in the superconducting phase. PACS numbers: 71.45.Lr, 71.20.Ps, 71.18.+y.