Crystalline phase identification for hafnium-based ferroelectrics by diffraction techniques has been elusive. We use density-functional-theory (DFT)-assisted extended X-ray absorption fine-structure spectroscopy (EXAFS) to determine the crystal symmetry of thin hafnium zirconium oxide (Hf0.46Zr0.54O2) films grown by atomic layer deposition. Ferroelectric switching in TiN/Hf0.46Zr0.54O2/TiN metal-insulator-metal capacitors is verified. Grazing-incidence fluorescence-yield mode Hf L3 and Zr K absorption edge EXAFS data are compared with reference data calculated from DFT-based atomic coordinates for various structural phases of Hf0.5Zr0.5O2. Via EXAFS multiphase fitting, we confirm that the frequently invoked polar orthorhombic Pca21 phase is present in ferroelectric hafnium zirconium oxide, along with an equal amount of the nonpolar monoclinic P21/c phase. For comparison, we verify that paraelectric HfO2 films exhibit the P21/c phase.