We consider a distributed system consisting of autonomous mobile computing entities, called robots, moving in a specified space. The robots are anonymous, oblivious, and have neither any access to the global coordinate system nor any explicit communication medium. Each robot observes the positions of other robots and moves in terms of its local coordinate system. To investigate the self-organization power of robot systems, formation problems in the two dimensional space (2D-space) have been extensively studied. Yamauchi et al. (DISC 2015) introduced robot systems in the three dimensional space (3D-space). While existing results for 3D-space assume that the robots agree on the handedness of their local coordinate systems, we remove the assumption and consider the robots without chirality. One of the most fundamental agreement problems in 3D-space is the plane formation problem that requires the robots to land on a common plane, that is not predefined. It has been shown that the solvability of the plane formation problem by robots with chirality is determined by the rotation symmetry of their initial local coordinate systems because the robots cannot break it. We show that when the robots lack chirality, the combination of rotation symmetry and reflection symmetry determines the solvability of the plane formation problem because a set of symmetric local coordinate systems without chirality is obtained by rotations and reflections. This richer symmetry results in the increase of unsolvable instances compared with robots with chirality and a flaw of existing plane formation algorithm. In this paper, we give a characterization of initial configurations from which the robots without chirality can form a plane and a new plane formation algorithm for solvable instances.