<p>Metamaterials with negative Poisson’s ratio and negative thermal expansion can be systematically designed. However, the question of how these two properties compete for negative values, if at all, remains understudied. To address this, in the frame of an enlarged design space, Pareto fronts are here approximated by considering a multiobjective topology optimization problem of bi-material microstructures. The proportions and spatial distribution of the two materials are non-trivial aspects when the relative importance of the design objectives changes. Consistently, a single and global volume constraint is used instead of enforcing individual material volume fractions. Besides the typical bulk-oriented stiffness constraint, this design framework also includes the less investigated shear-oriented stiffness case. Different material symmetries and initial designs are explored, such that the results include a variety of known auxetics for validation. However, when widening the scope to thermoauxeticity, most of the bi-material designs found here, emerging in a Pareto sense, reveal novel and yet non-reported design transitions. The two base materials proportions change on the top of a layout that reasonably changes for required functionality. Additionally, the analysis of scale-size effects is here extended to thermoelasticity to conclude that the thermoelastic coefficients also show a good convergence towards the homogenization predictions. Overall, this work offers key insights into the trade-offs that must take place when designing metamaterials for double-negative indexes.</p>