activity. Examples of known natural gyrotropic systems are quartz [@Arago], some organic liquids and aqueous solutions of sugar and tartaric acid [@Melrose], the Pb$_5$Ge$_3$O$_{11}$ compound [@experiment; @Koiiak], and the layered crystal (C$_5$H$_{11}$NH$_3$)$_2$ZnCl$_4$ [@Pnma-P212121]. Finding novel natural gyrotropic materials has great fundamental interest. It may also lead to the design of novel devices, such as optical circulators and amplifiers, especially if the [*sign*]{} of the optical rotation can be efficiently controlled by an external factor that is easy to manipulate.

When searching for new natural gyrotropic materials, one should remember the observation of Pasteur that chiral crystals display spontaneous optical activity, which reverses sign when going from the original structure to its mirror image [@Pasteur]. Hence it is worthwhile to consider a newly discovered class of materials that are potentially chiral, and therefore may be naturally gyrotropic. This class is formed by electrotoroidic compounds (also called ferrotoroidics [@Shmid]). These are systems that possess an electrical toroidal moment, or equivalently, exhibit electric vortices [@Dubovik]. Such intriguing compounds were predicted to exist around nine years ago [@Ivan], and were found experimentally only recently [@Gruverman-Scott-vortex; @Balke-Kalinin; @Vasudevan; @Nelson; @Gregg]. One may therefore wonder if this new class of materials is indeed naturally gyrotropic, and/or 