Abstract We present resolved Herschel images of circumbinary debris discs in the α CrB (HD 139006) and β Tri (HD13161) systems. By modelling their structure, we find that both discs are consistent with being aligned with the binary orbital planes. Though secular perturbations from the binary can bring the disc into alignment, in both cases the alignment time at the distances at which the disc is resolved is greater than the stellar age, so we conclude that the coplanarity was primordial. Neither disc can be modelled as a narrow ring, requiring extended radial distributions. To satisfy both the Herschel and mid-infrared images of the α CrB disc, we construct a model that extends from 1 to 300 au, whose radial profile is broadly consistent with a picture where planetesimal collisions are excited by secular perturbations from the binary. However, this model is also consistent with stirring by other mechanisms, such as the formation of Pluto-sized objects. The β Tri disc is modelled as a disc that extends from 50 to 400 au. A model with depleted (rather than empty) inner regions also reproduces the observations and is consistent with binary and other stirring mechanisms. As part of the modelling process, we find that the Herschel PACS (Photodetector and Array Camera & Spectrometer) beam varies by as much as 10 per cent at 70 μm and a few per cent at 100 μm. The 70 μm variation can therefore hinder image interpretation, particularly for poorly resolved objects. The number of systems in which circumbinary debris disc orientations have been compared with the binary plane is now 4. More systems are needed, but a picture in which discs around very close binaries (α CrB, β Tri and HD 98800, with periods of a few weeks to a year) are aligned, and discs around wider binaries (99 Her, with a period of 50 yr) are misaligned, may be emerging. This picture is qualitatively consistent with the expectation that the protoplanetary discs from which the debris emerged are more likely to be aligned if their binaries have shorter periods.