The description of electron correlation in quantum chemistry often relies on multi-index quantities. Here, we examine a compressed representation of the long-range part of electron correlation that is associated with dispersion interactions. For this purpose, we perform coupled-cluster singles and doubles (CCSD) computations on localized orbitals, and then extract the portion of CCSD amplitudes corresponding to dispersion energies. Using singular value decomposition, we uncover that a very compressed representation of the amplitudes is possible in terms of occupied-virtual geminal pairs located on each monomer. These geminals provide an accurate description of dispersion energies at medium and long distances. The corresponding virtual orbitals are examined by further singular value decompositions of the geminals. We connect each component of the virtual space to the multipole expansion of dispersion energies. Our results are robust with respect to basis set change and hold for systems as large as the benzene-methane dimer. This compressed representation of dispersion energies paves the way to practical and accurate approximations for dispersion, for example, in local correlation methods.