Heterochiral self-replication of loop structures in cellular automata spaces cannot be achieved by pooling a state-transition function supporting self-replication with its corresponding mirror-transformed transition function because some chiral rules specific to right-handed replication are contradicted by chiral rules specific for left-handed replication. In a subsequent study, a less-strict functional heterochirality of self-replication, notionally achievable by application of state-set permutation-transformation to structures and state-transition functions was investigated. Although this strategy reduces the number of rule contradictions between left-handed and right-handed self-replication, state-set permutations applied to left-handed replication do not enable functional coexistence of left- and right-handed self-replication. This work describes the consequences of rewriting von Neumann state-transition rules as a larger set of corresponding Moore state-transition rules. It was observed that there are less state-transition rule contradictions between left- and right-handed self-replication under the Moore rules state-transition function. The observations that state-set permutation-transformations and Moore state-transition rules independently reduce rule contradictions between left- and right-handed self-replication suggested the hypothesis that both strategies implemented together may reduce rule contradictions between left- and right-handed self-replication to zero, enabling functional coexistence of left- and right-handed self-replication. This work reports success in implementing functional coexistence of left- and right-handed self-replication of the J. Byl structure (1989) by application of a Moore rules state-transition function, and any one of six state-set permutation transformations applied to left-handed self-replication.