G rowing evidence indicates that the superconducting pyrochlore Cd 2 Re 2 O 7 exhibits a structural phase transition at T c = 200 K with an unusual tensor character 1-3 . The structural order parameter for this state is two-dimensional, and spanned by distinct but nearly degenerate crystallographic structures I4 1 22 and I4m2 (ref. 1). Symmetry rules imply that the low-energy excitations of the ordered state are Goldstone phonons, or long wavelength fluctuations between the two crystal structures. These are the structural equivalents of magnons in an XY antiferromagnet, with the two crystal structures analogous to orthogonal spin directions in the xy-plane. Goldstone phonons have been observed in Raman spectroscopy 3 , but high-resolution X-ray and neutron scattering experiments have produced conflicting assignments of the static low-temperature structure 4-6 . Here, we use optical secondharmonic generation with polarization sensitivity to assign the I4m2 structure unambiguously and verify an auxiliary condition on the structure that is implied by the order parameter symmetry. We also show that the temperature dependence of the order parameter is consistent with thermal occupation of the Goldstone mode. The methodology may be applied more widely in characterizing ordered states in matter.Various experimental probes show a continuous cubic-totetragonal transition in Cd 2 Re 2 O 7 at T c (refs 1,2,4-10). However, as Anderson and Blount pointed out over 40 years ago 11 , a less conventional order parameter, possibly with ferroelectric character, must accompany strain to make any cubic-to-tetragonal transition continuous. Experiments have indeed ruled out strain as the primary order parameter for Cd 2 Re 2 O 7 (ref. 2), and both X-ray and neutron diffraction reveal broken inversion symmetry below T c (refs 4-6). Theoretical analysis indicates that the true order parameter is a second-rank pseudotensor, corresponding to the E u representation of the cubic point group 1,2 , shown in Fig. 1. The more familiar types of order all have lower rank: vectors for ferroelectricity, inversion-symmetric pseudovectors for ferromagnetism and second-rank tensors for ferroelasticity. Like the vector order of ferroelectricity, E u tensor order in In 0 Out