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We report anisotropic resistivity measurements of the heavy-fermion superconductor CeRh$_2$As$_2$ in magnetic fields up to 16 T and temperatures down to 0.35 K. The measured CeRh$_2$As$_2$ resistivity shows a signature corresponding to the suggested quadrupole density wave order state at $T_0 \sim$ 0.5 K for both measured directions. For a magnetic field applied along the tetragonal $a$ axis, $T_0$ is enhanced with magnetic field reaching $\sim$1.75 K at 16 T. Further, a magnetic field-induced transition occurs at $B_m \sim $ 8.1 T corresponding to a change to a new broken symmetry state. For a magnetic field applied along the $c$ axis, $T_0$ is suppressed below our base temperature $\sim$0.35 K by $B \sim$ 4.5 T, a field close to the previously reported field-induced transition within the superconducting state suggested to be from an even-parity to an odd-parity state. Our results indicate that the multiple superconducting phases in CeRh$_2$As$_2$ are intimately tied to the suppression of the proposed quadrupole-density-wave phase at $T_0$., Comment: 6 pages and 5 figures