The pygmy dipole resonance has been studied in the proton-magic nucleus 124Sn with the (α, α'γ) coincidence method at Eα=136 MeV. The comparison with results of photon-scattering experiments reveals a splitting into two components with different structure: one group of states which is excited in (α, α'γ) as well as in (γ, γ') reactions and a group of states at higher energies which is only excited in (γ, γ') reactions. Calculations with the self-consistent relativistic quasiparticle time-blocking approximation and the quasiparticle phonon model are in qualitative agreement with the experimental results and predict a low-lying isoscalar component dominated by neutron-skin oscillations and a higher-lying more isovector component on the tail of the giant dipole resonance.