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We exploit fluctuational electrodynamics to present trace expressions for thetorque experienced by arbitrary objects in a passive, nonabsorbing,rotationally invariant background environment. We present trace expressions forequilibrium Casimir torque which complement recently derived nonequilibriumtorque expressions and explicate their relation to the Casimir free energy. Wethen use the derived trace expressions to calculate, via Lagrange duality,semianalytic structure-agnostic bounds on the Casimir torque between ananisotropic (reciprocal or nonreciprocal) dipolar particle and a macroscopicbody composed of a local isotropic electric susceptibility, separated byvacuum.

Trace expressions and associated limits for equilibrium Casimir torque