We analyze the origin and properties of the chaotic dynamics of two atomic ensembles in a driven-dissipative experimental setup, where they are collectively damped by a bad cavity mode and incoherently pumped by a Raman laser. Starting from the mean-field equations, we explain the emergence of chaos by way of quasiperiodicity -- presence of two or more incommensurate frequencies. This is known as the Ruelle-Takens-Newhouse route to chaos. The equations of motion have a $mathbb{Z}_{2}$-symmetry with respect to the interchange of the two ensembles. However, some of the attractors of these equations spontaneously break this symmetry. To understand the emergence and subsequent properties of various attractors, we concurrently study the mean-field trajectories, Poincar{e} sections, maximum and conditional Lyapunov exponents, and power spectra. Using Floquet analysis, we show that quasiperiodicity is born out of non $mathbb{Z}_{2}$-symmetric oscillations via a supercritical Neimark-Sacker bifurcation. Changing the detuning between the level spacings in the two ensembles and the repump rate results in the synchronization of the two chaotic ensembles. In this regime, the chaotic intensity fluctuations of the light radiated by the two ensembles are identical. Identifying the synchronization manifold, we understand the origin of synchronized chaos as a tangent bifurcation intermittency of the $mathbb{Z}_{2}$-symmetric oscillations. At its birth, synchronized chaos is unstable. The interaction of this attractor with other attractors causes on-off intermittency until the synchronization manifold becomes sufficiently attractive. We also show coexistence of different phases in small pockets near the boundaries.
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