Magnetic fields are an important but largely unknown ingredient of planetary nebulae. They have been detected in oxygen-rich asymptotic giant branch (AGB) and post-AGB stars, and may play a role in the shaping of their nebulae. Here we present SCUBA submillimetre polarimetric observations of four bipolar planetary nebulae and post-AGB stars, including two oxygen-rich and two carbon-rich nebulae, to determine the geometry of the magnetic field by dust alignment. Three of the four sources (NGC 7027, 6537 and 6302) present a well-defined toroidal magnetic field oriented along their equatorial torus or disc. NGC 6302 may also show field lines along the bipolar outflow. CRL 2688 shows a complex field structure, where part of the field aligns with the torus, whilst an other part approximately aligns with the polar outflow. It also presents marked asymmetries in its magnetic structure. NGC 7027 shows evidence for a disorganized field in the south-west corner, where the SCUBA shows an indication for an outflow. The findings show a clear correlation between field orientation and nebular structure.