  We examine the role of charge, structural, and spin degrees of freedom in the
previously poorly understood phase transition in the 5$d^1$ transition-metal
double perovskite Cs$_2$TaCl$_6$, using a combination of computational and
experimental techniques. Our heat capacity measurements of single-crystalline
Cs$_2$TaCl$_6$, reveal a clear anomaly at the transition temperature,
$T_\mathrm{Q}$, 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
Cs$_2$TaCl$_6$, single crystals lend support to our calculations. Our work
provides new insight into the phase transition in Cs$_2$TaCl$_6$, at
$T_\mathrm{Q}$, and demonstrates the utility of this combination of techniques
in understanding the complex physics of hidden orders in paramagnetic
spin-orbit-entangled compounds.
