The excitations responsible for producing high-temperature superconductivity in the copper oxides have yet to be identified. Two promising candidates are collective spin excitations and phonons(1). A recent argument against spin excitations is based on their inability to explain structures observed in electronic spectroscopies such as photoemission(2-5) and optical conductivity(6,7). Here, we use inelastic neutron scattering to demonstrate that collective spin excitations in optimally doped La2-xSrxCuO4 are more structured than previously thought. The excitations have a two-component structure with a low-frequency component strongest around 18 meV and a broader component peaking near 40 - 70 meV. The second component carries most of the spectral weight and its energy matches structures observed in photoemission(2-5) in the range 50 - 90 meV. Our results demonstrate that collective spin excitations can explain features of electronic spectroscopies and are therefore likely to be strongly coupled to the electron quasiparticles.