We theoretically and experimentally investigate low-Reynolds-number propulsion of geometrically achiral planar objects that possess a dipole moment and that are driven by a rotating magnetic field. Symmetry considerations (involving parity, $\widehat{P}$, and charge conjugation, $\widehat{C}$) establish correspondence between propulsive states depending on orientation of the dipolar moment. Although basic symmetry arguments do not forbid individual symmetric objects to efficiently propel due to spontaneous symmetry breaking, they suggest that the average ensemble velocity vanishes. Some additional arguments show, however, that highly symmetrical ($\widehat{P}$-even) objects exhibit no net propulsion while individual less symmetrical ($\widehat{C}\widehat{P}$-even) propellers do propel. Particular magnetization orientation, rendering the shape $\widehat{C}\widehat{P}$-odd, yields unidirectional motion typically associated with chiral structures, such as helices. If instead of a structure with a permanent dipole we consider a polarizable object, some of the arguments have to be modified. For instance, we demonstrate a truly achiral ($\widehat{P}$- and $\widehat{C}\widehat{P}$-even) planar shape with an induced electric dipole that can propel by electro-rotation. We thereby show that chirality is not essential for propulsion due to rotation-translation coupling at low Reynolds number.