The Cosmological Principle assumes a statistically isotropic Universe, but the Cosmic Microwave Background (CMB) exhibits some anomalous statistical features, such as the hemispherical power asymmetry, that challenge this core assumption. We aim to expand the characterization of this anomaly by investigating the uniformity of the CMB's morphological and topological properties, testing whether the asymmetry extends beyond the variance of the temperature field. We evaluate the three Minkowski Functionals (MFs) on local patches of the \textit{Planck} SMICA temperature map and compare them against realistic simulations. By fitting the local MFs to the analytical expectations for Gaussian isotropic fields, we extract local estimates of the temperature variance, the variance of the field gradients, and the goodness-of-fit. We then evaluate the amplitude and alignment of dipoles in these quantities. We confirm the highly significant variance dipole (p-value $\sim0.3\%-1.0\%$). Furthermore, we report a moderately significant dipole in the gradient variance (p-value $\sim2.6\%-3.7\%$) that is statistically independent of the previous dipole under the $Λ$CDM model. We also find a mild spatial variation in the goodness-of-fit to the Gaussian isotropic predictions (p-value $\sim2.1\%-5.6\%$). Remarkably, the \textit{Planck} dipoles for all three quantities point toward the same region of the sky. We find that the all asymmetries are well described by dipoles. The Hemispherical Asymmetry in the CMB extends beyond its local variance, as we found it to be also present in its local morphology, with a possible hint of non-Gaussianity. These results provide a more complete characterization of the Hemispherical Asymmetry, and will therefore contribute to better determine the nature of the physical effect behind it, whether cosmological, residual foregrounds, or unknown systematics. (Abridged)