High-resolution X-ray and neutron powder diffraction are used to reveal details of the spin-reorientation transition in the layered oxide pnictide CeMnAsO. Above 38 K, the localized moments on Mn2+ are antiferromagnetically ordered in a checkerboard fashion within the antifluorite-type MnAs planes and are oriented perpendicular to the planes. Below 38 K, reorientation of these moments into the planes commences. This is complete by 34 K and is coincident with long-range ordering of the Ce3+ moments. The Ce3+ and Mn2+ moments have an arrangement that is different in detail from that in the isostructural NdMnAsO and PrMnSbO. There is no evidence for structural distortion, as found for PrMnSbO and related Pr3+-containing compounds, although there is evidence for a very slight (0.025%) misfit between the magnetic and structural cells below the spin-reorientation transition. It is clarified that neutron powder diffraction methods are unable to distinguish between collinear and noncollinear arrangements of manganese and lanthanide moments when the moments have a component parallel to the MnAs planes. A proposal from computational analysis that NdMnAsO and CeMnAsO should adopt different magnetic structures on the basis of the different balances between biquadratic and antisymmetric exchange interactions should be tested using alternative methods.