We report on the experimental observations of chaos synchronizations among orthogonally polarized emissions in a dual-polarization laser. With a three-mode scheme with a single mode in one polarization and two modes in orthogonal polarizations, synchronization was achieved by the cross-saturation dynamics of population inversions when one of the dual-polarized emissions was subjected to external perturbations, i.e., self-mixing modulation or optical fiber feedback. In-phase synchronization or lag synchronization among orthogonally polarized emissions was achieved depending on the degree of cross saturation in the self-mixing modulation. Synchronization of random bursting was observed in the fiber feedback, in which two chaotic-spiking modes in one polarization with anticorrelated intensity variations synchronize the remaining mode in the orthogonal polarization cooperatively with their total intensity variation. Information sender-mediator-receiver relationships among modes, which represent the dynamical roles of individual modes for establishing the observed three types of collective synchronizations, were identified in terms of an information circulation analysis.