Figure A1.

(a) Additional full diffraction scans taken at room temperature and with varying incident beam energy $E_i$ further illustrate the spin-allowed CDW periodicity of Bi${}_2$Te${}_2$Se(111) along $\overline{\mathsf{\Gamma }\mathrm{M}}$. The corresponding satellite peaks (grey dashed vertical lines) appear next to the specular and first-order diffraction peaks (grey dash-dotted vertical lines). (b) An additional full diffraction scan (logarithmic scale), with the sample cooled down, illustrates that the asymmetry between $+{\mathbf{g}}_{\mathrm{CDW}}$ and $-{\mathbf{g}}_{\mathrm{CDW}}$, as described in the main text, is also evident for the satellite peaks next to the first-order diffraction peaks.