A figure is achiral with an orthogonal matrix  and a vector . The determinant of  is either 1 or &minus;1 then. If it is &minus;1 the isometry is orientation-reversing, otherwise it is orientation-preserving.
A general definition of chirality based on group theory exists. It does not refer to any orientation concept: an isometry is direct if and only if it is a product of squares of isometries, and if not, it is an indirect isometry. The resulting chirality definition works in spacetime.