Polarization-resolved coherent anti-Stokes Raman scattering (CARS) is usually applied to measure the depolarization ratio in solutions or evidence orientation effects in anisotropic media. We present an extensive approach based on multiple-field polarization-resolved CARS, in order to unravel the complexity of vibrational resonances up to the fourth-order symmetry, at the microscopic scale in nonisotropic media. The CARS polarized signals measured under a continuous variation of the incident pump and/or Stokes excitation beams are analyzed using a full tensorial picture both in the nonresonant and resonant regimes. This method evidences the strong influence of vibrational symmetries on polarized CARS, and more specifically the occurrence of Kleinman symmetry deviations at the vicinity of the Raman lines frequencies. This technique, illustrated on a cubic symmetry crystal, is general and can be applied to other medium symmetries.