Second harmonic generation (SHG) on the pyrochlore metal Cd2Re2O7 indicates the presence of three order parameters setting in below an inversion breaking transition. Here, we explore a possible structural explanation and relate it not only to the SHG data, but also to neutron and xray diffraction, where we find that such a structural scenario can explain certain reflection extinctions observed in single crystal x-ray data. From this analysis, we suggest future experiments that could be done to resolve this matter. Finally, we comment on the Landau-violating nature of the inversion breaking transition and its relation to similar phenomena observed in improper ferroelectrics.Although most pyrochlore oxides are insulating, Cd 2 Re 2 O 7 is a notable exception. It is a good metal and exhibits three phase transitions, one near 200 K (T s1 ), another near 120 K (T s2 ), and finally a low temperature superconducting transition (T c ) [1]. Structural data support the presence of inversion breaking at T s1 from a high temperature cubic phase (Fd3m) to a tetragonal phase (I4m2), whereas T s2 is claimed to be a weakly first-order transition to I4 1 22 [2]. There are several issues, though.First, x-ray [3] and neutron [4] data indicate very different atom displacements, these measurements being hampered by the fact that the structural distortion is weak. Second, single crystal x-ray diffraction [5] find reflection extinction conditions that are not consistent with I4m2. Finally, second harmonic generation (SHG) data [6] are consistent with I4m2, but are not consistent with its conversion to I4 1 22 below T s2 .Subsequently, Liang Fu suggested the possibility that a novel spin nematic phase might set in below T s1 [7], motivated by the fact that large changes occur in the resistivity and susceptibility at T s1 despite the weak nature of the structural distortions. This it turn motivated new SHG experiments [8] that supported this picture, in that they found that the previously observed SHG signal was secondary in nature in regards to its temperature dependence (that is, it varied as T s1 − T ). They found a new (but much smaller) signal that had a primary order parameter dependence ( √ T s1 − T ). From the azimuthal and polarization dependence of the SHG signal, the secondary one (as before) was identified as having E u symmetry relative to the high temperature cubic phase (consistent with I4m2), but the primary one was identified as having T 2u symmetry. Given the known temperature dependences of the two SHG signals, this implies the presence of a third-order term in the Landau free energy. Since a product of three odd parity order parameters is not allowed, this indicates the presence of a third order parameter of even parity that also has a primary temperature dependence, identified as T 1g . These two primary order parameters were speculated to be associated with spin nematic order, with the secondary order parameter being a weak structural effect driven by the primary or-der. The three together define a trilinear term in...