This term is frequently used in chemistry context: There chirality exists for example for a molecule which does not possess an asymmetric chiral carbon atom but perpendicular disymmetric planes due to restricted rotation around a chemical bond in the molecule. Planar chirality is most commonly encountered in compounds like (E)-cyclooctene, some di-substituted ferrocenes, and certain monosubstituted paracyclophanes. Nature rarely provides planar chiral molecules; cavicularin being an exception.
However, planar chirality is basically a mathematical term, finding use in natural sciences like areas of physics, namely astronomy, optics and metamaterials. Recent occurrences in latter two fields are dominated by microwave applications and micro- and nanostructured planar interfaces for visible and infrared light.