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This Letter reports on the first measurements of transverse momentum dependent flow angle Ψn and flow magnitude vn fluctuations, determined using new four-particle correlators. The measurements are performed for various centralities in Pb-Pb collisions at a centre-of-mass energy per nucleon pair of sNN−−−√ = 5.02 TeV with ALICE at the CERN Large Hadron Collider. Both flow angle and flow magnitude fluctuations are observed in the presented centrality ranges and are strongest in the most central collisions and for a transverse momentum pT>2 GeV/c. Comparison with theoretical models, including iEBE-VISHNU, MUSIC, and AMPT, show that the measurements exhibit unique sensitivities to the initial state of heavy-ion collisions.