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Mode-locked lasers play the role of the ideal testbeds for studying self-coherent structures, dissipative solitons, with stable spatiotemporal profiles supported by the balance between dispersion and nonlinearity. However, under some conditions, the profile and energy of the solitons can oscillate (breath) periodically. Unlike the previous studies of different scalar mechanisms of breathers emergence, we demonstrate experimentally and theoretically a new vector mechanism where breathing of the output power is a result of the heteroclinic dynamics. The heteroclinic orbit is a trajectory periodically evolving nearby the neighborhood of one of the orthogonal states of polarization with further switching to and evolving nearby the other state of polarization. The dwelling time for the trajectory near each orthogonal state is determined by the cavity anisotropy adjustable with the help of the polarization controller for the pump wave. The obtained results on tunability of breathing dynamics enables unlocking novel techniques of the flexible control and manipulation of the polarization dynamics of dissipative solitons.