To elucidate the spin dynamics of La2-xSrxCuO4, which couples with the charge degree of freedom, the spin excitations spanning the characteristic energy (Ecross~ 35-40 meV) are investigated for underdoped x = 0.10 and optimally doped (OP) x = 0.16 through inelastic neutron scattering measurements. Analysis based on a two-component picture of high-quality data clarified the possible coexistence of upright standing incommensurate (IC) excitations with gapless commensurate (C) excitations in the energy-momentum space. The IC component weakens the intensity toward Ecrosswith increasing energy transfer (hω), whereas the C component strengthens at high-hω regions. The analysis results imply that the superposition of two components with a particular intensity balance forms an hourglass-shaped excitation in appearance. Furthermore, the temperature dependence of the intensity of each component exhibits different behaviors; the IC component disappears near the pseudo-gap temperature (Tz.ast;) upon warming, whereas the C component is robust against temperature. These results suggest the itinerant electron spin nature and the localized magnetism of the parent La2CuO4in IC and C excitations, respectively, similar to the spin excitations in the OP YBa2Cu3O6+δand HgBa2CuO4+δwith higher superconducting-transition temperatures.