Experimental data are presented which demonstrate the existence of a fine structure in extended X-ray absorption spectra due to interference effects in the initial photon state (piXAFS). Interference occurs between the incident electromagnetic wave and its coherently scattered waves from neighboring atoms. Using fine platinum and tungsten powders as well as polycrystalline platinum foil, piXAFS was measured in high-precision absorption experiments at beamline X1 at HASYLAB/DESY over a wide energy range. piXAFS is observed below and above absorption-edge positions in both transmission and total-electron-yield detection. Based on experimental data it is shown that piXAFS is sensitive to geometric atomic structure. Fourier-transformed piXAFS data carry information, comparable with that of EXAFS, about the short-range-order structure of the sample. Sharp structures occur in piXAFS when a Bragg backscattering condition of the incident X-rays is fulfilled. They allow precise measurement of long-range-order structural information. Measured data are compared with simulations based on piXAFS theory. Although piXAFS structures are similarly observed in two detection techniques, the importance of scattering off the sample for the measurements needs to be investigated further. Disentangling piXAFS, multielectron photoexcitations and atomic XAFS in high-precision measurements close to absorption edges poses a challenge for future studies.