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New sideband partners of the yrast bands built on the pi(h11/2)nu(h11/2) configuration were identified in 55Cs, 57La, and 61Pm N = 75 isotones of 134Pr. These bands form with 134Pr unique doublet-band systematics suggesting a common basis. Aplanar solutions of 3D tilted axis cranking calculations for triaxial shapes define left- and right-handed chiral systems out of the three angular momenta provided by the valence particles and the core rotation, which leads to spontaneous chiral symmetry breaking and the doublet bands. Small energy differences between the doublet bands suggest collective chiral vibrations.