The fully differential cross sections for the electron-impact ionization of atomic targets, prepared in a given circular state using laser-pumping, reveal a dependence on the inversion of the helicity of the exciting photon. This 'dichroism' effect was shown to be strongly dependent on the geometrical arrangements of the experiment both theoretically and experimentally. In addition, as shown theoretically in this paper, the dichroism may also vanish at certain 'non-geometrical' points that can be deduced analytically within the first Born approximation. More elaborate calculations using the distorted-wave Born approximation confirm this analysis. On the basis of this study, we further suggest a possible explanation for a structure observed recently in the state-resolved fully differential cross sections for a sodium target.