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We describe a mechanism by which the longitudinal thermal conductivity κ_{xx}, measured in an in-plane magnetic field, oscillates as a function of field angle in layered nodal superconductors. These oscillations occur when the spin-orbit splitting at the nodes is larger than the nodal scattering rate, and are complementary to vortex-induced oscillations identified previously. In sufficiently anisotropic materials, the spin-orbit mechanism may be dominant. As a particular application, we focus on the cuprate high-temperature superconductor YBa_{2}Cu_{3}O_{6+x}. This material belongs to the class of Rashba bilayers, in which individual CuO_{2} layers lack inversion symmetry although the crystal itself is globally centrosymmetric. We show that spin-orbit coupling endows κ_{xx}/T with a characteristic dependence on magnetic field angle that should be easily detected experimentally, and argue that for underdoped samples the spin-orbit contribution is larger than the vortex contribution. A key advantage of the magnetothermal conductivity is that it is a bulk probe of spin-orbit physics, and therefore not sensitive to inversion breaking at surfaces.