We uncover two anomalous features in the nonlocal transport behavior of two-dimensional metallic materials with spin-orbit coupling. Firstly, the nonlocal resistance can have negative values and oscillate with distance, even in the absence of a magnetic field. Secondly, the oscillations of the nonlocal resistance under an applied in-plane magnetic field (Hanle effect) can be asymmetric under field reversal. Both features are produced by direct magnetoelectric coupling, which is possible in materials with broken inversion symmetry but was not included in previous spin diffusion theories of nonlocal transport. These effects can be used to identify the relative contributions of different spin-charge conversion mechanisms. They should be observable in adatom-functionalized graphene, and may provide the reason for discrepancies in recent nonlocal transport experiments on graphene.Introduction-The ability to convert between macroscopic spin and charge degrees of freedom is a distinctive feature of materials with spin-orbit coupling (SOC), and has fundamental importance for spintronics research [1]. The two most prominent examples of spin-charge conversion are the spin Hall effect (SHE) [2, 3] and currentinduced spin polarization (CISP) [4,5]: when an electric current J is injected into a material with SOC, it can generate a spin current J (the SHE), and/or a nonequilibrium spin polarization m (CISP). Spin-charge conversion can be detected and studied using nonlocal transport experiments [6-9], a well-established technique that has been applied to two-dimensional (2D) quantum spin Hall insulators [10], 3D topological Kondo insulators [11], and many other systems [12][13][14][15]. These experiments rely on a combination of spin-charge conversion processes: when J is injected at one position, the SHE (CISP) converts part of it to J (m), which diffuses across the device, and is then converted back into J by the inverse SHE (inverse CISP) and measured via the nonlocal electrical resistance R nl . Moreover, applying an in-plane magnetic field induces Hanle precession, which is observed as an oscillation of R nl with distance and field strength [16].