The statistical relationships among the turbulence structures of the streamwise velocity fluctuations along the streamwise and azimuthal directions in a turbulent pipe flow were examined using direct numerical simulation data at Re-tau = 3008. Two-point correlations of the streamwise velocity fluctuations showed a linear relationship between the streamwise and azimuthal length scales (l(x) and l(theta)), where l(theta)/l(x) = 0.07 along the wall-normal distance, indicating the long coherent structures called very-large-scale motions (VLSMs). The one-dimensional pre-multiplied energy spectra of the streamwise velocity fluctuations showed that the streamwise and the azimuthal wavelengths (lambda(x) and lambda(theta)) grew linearly along the wall-normal distance, lambda(x)/y = 20 and lambda(theta)/y = 7, respectively. The ratio between the two linear relationships was determined to be lambda(theta)/lambda(x) = 0.35, indicative of large-scale motions (LSMs). The energetic modes obtained from a proper orthogonal decomposition (POD) analysis using the translational invariance method showed that the averaged helical angles of the wall mode (i(x) < i(theta); beta < 0.1 rad, where i(x) and i(theta) are the streamwise and azimuthal mode numbers and beta is the helical angle) and lift mode (i(x) >= i(theta); beta >= 0.1 rad) were related to l(theta)/l(x) = 0.07 (VLSMs) and lambda(theta)/lambda(x) approximate to 0.35 (LSMs), respectively. The superposition of the energetic POD modes showed the superimposed X-shaped patterns. The helical angle of the wall mode in the near-wall region was similar to that in the outer region, implying the existence of the VLSMs in the entire wall-normal distance. The LSMs showed more inclined X-shaped patterns. The LSMs were concatenated with the azimuthal offsets to form meandering VLSMs. Most of the VLSMs and LSMs in the near-wall region inclined smaller and larger than 10 degrees (0.17 rad), respectively. In the core region, VLSMs were distributed more helically along the azimuthal direction due to the space limitations of the pipe geometry. Published by AIP Publishing.