Abstract: Recently, a process has been proposed for generating negatively-curved carbon schwarzites via zeolite-templating (Braun et al., 2018). However, the proposed process leads to atomistic models which are not very symmetric and often rather defective. In the present work, an improved generation approach is developed, by imposing symmetry constraints, which systematically leads to defect-free, hence more stable, schwarzites. The stability of the newly predicted symmetric schwarzites is also compared to that of other carbon nanostructures (in particular carbon nanotubes – CNTs), which could also be accommodated within the same templates. Our results suggest that only a few of these (such as FAU, SBT and SBS) can fit schwarzites more stable than CNTs. Our predictions could help experimentalists in the crucial choice of the template for the challenging synthesis of schwarzites. Furthermore, being highly symmetric and stable phases, the models could also be synthesized by means of other experimental procedures.