Lattice-based mechanical metamaterials can be tailored for a wide variety of applications by modifying the underlying mesostructure. However, most existing lattice patterns take symmetry as a starting point. We show that asymmetric lattice patterns can be more likely to have certain mechanical properties than symmetric lattice patterns. To directly compare the effects of asymmetric versus symmetric lattice arrangements, a constrained design space is defined. A generative design process is used to generate both symmetric and asymmetric lattice patterns within the design space. Asymmetric lattice patterns are shown to have distinct metamaterial property spaces from symmetric lattice patterns. Key design features are identified that are present predominantly in asymmetric lattice patterns. We show that asymmetric lattice patterns with two of these features (arrows and spider nodes) are more likely to induce a broader range of Poisson’s ratios and larger shear stiffness values, respectively, compared to lattice patterns without these features. In addition, we show that symmetry can play a role in hampering the impact of multiple features when present. We extend this work to the development of heuristic functions for metamaterial design optimization. These heuristics are shown to reduce the number of function evaluations required when optimizing metamaterials via an evolutionary design process. This work provides insights into the benefits of using asymmetric lattice patterns in select metamaterial design applications.