Proper generation of atoms by symmetry requires that atom coordinates be consistent with the orientation of symmetry elements. That is, the people who publish or otherwise report atom coordinates have to assume a particular geometric relationship between the symmetry elements and the coordinate axes in terms of which the atomic positions are given. In some point and space groups there is only one reasonable choice for the mutual orientation of coordinate axes and symmetry, but in some groups there may be two or more possibilities and ambiguity may arise.
Crystals. The preferred symmetry option for crystals is normally Space-group, which reads the symmetry operations from a selection of those in the Tables. The main part of the Tables themselves (section 4.3 of Volume I, pre-1983 or Chapter 6 of Volume A, post-1983) lists only some of the possible orientations of space-group symmetry for monoclinic and orthorhombic crystals, but the alternate orientations can be generated by ATOMS, as described in the detailed instructions for the Space-group option. The alternate settings are listed in Table 6.2.1 of Volume 1 or Table 4.3.1 of Volume A of the Tables.
A consistent orientation of symmetry elements with respect to axes is not always used for space groups in the International Tables for X-ray Crystallography. As an example of alternate orientations, consider the point group C3v. The two simplest orientations for the vertical mirror planes are either parallel to the a1 and a2 axes, or in between, rotated 30 degrees. These two orientations are embodied in the two space groups P31m-C3v2 (no. 157) and P3m-C3v1 (no. 156). If the same orientation were used for both, one of them would have a C-centered Bravais lattice, the avoidance of which the compilers of the Tables apparently considered more important than consistent axial orientation.
It is important to realize that these different possibilities exist, especially when using face or form indices from textbooks or the literature; it may be necessary to transform the indices to conform with the orientation used in the structure description.
Molecules. Using the Point Group symmetry option, which derives the symmetry operations from space-group listings in the Tables, unavoidably makes certain assumptions about orientation. In monoclinic symmetry (point groups 2-C2, m-Cs, and 2/m-C2h) the unique axis is taken to be parallel to c or z, rather than b or y (as in crystals). In the orthorhombic point group mm2-C2v the two-fold axis is assumed to be parallel to c. If the atomic coordinates for the molecule are given in terms of a non-standard setting, an alternate to manual rotation of the coordinates before input is to use the Space-group option, specifying a space group in the point group of interest which has the alternate orientation. Some examples of alternate settings related by 30 and 45 degree rotations on the c axis are given in the section on the Space-group option.