We examine the role of charge, structural, and spin degrees of freedom in the previously poorly understood phase transition in the 5d1 transition-metal double perovskite Cs2TaCl6 using a combination of computational and experimental techniques. Our heat capacity measurements of single-crystalline Cs2TaCl6 reveal a clear anomaly at the transition temperature, TQ, which was not previously observed in polycrystalline samples. Density functional calculations indicate the emergence of local charge quadrupoles in the cubic phase, mediated by the paramagnetic spins or local structural distortions which then develop into long-range-ordered charge quadrupoles in the tetragonal phase. Our resonant elastic x-ray scattering on Cs2TaCl6 single crystals lends support to our calculations. Our work provides insight into the phase transition in Cs2TaCl6 at TQ, and demonstrates the utility of this unique combination of techniques in understanding the complex physics of hidden orders in paramagnetic spin-orbit-entangled compounds.