We introduce and study two classes of neutrosophic continuous mappings: the neutrosophic irresolute $\delta$-$\beta^*$-continuous mappings (NIr $\delta$-$\beta^*$ CM) and the $\delta$-$\beta^*$-neutrosophic contra $\delta$-$\beta^*$-continuous mappings (NC $\delta$-$\beta^*$ CM). We establish their fundamental properties and provide characterizations in terms of preimages of $\delta$-$\beta^*$-open and $\delta$-$\beta^*$-closed sets. The role of each notion related to the other is shown and analyzed through implication chains, (non-)equivalences under mild hypothesis and stability result under composition, subspaces, and products. Then an extended framework to the $m$-polar setting is shown; in particular, the definitions of the $m$-polar neutrosophic irresolute $\delta$-$\beta^*$-continuous mappings (MPNIr $\delta$-$\beta^*$ CM) and $m$-polar neutrosophic contra $\delta$-$\beta^*$-continuous mappings (MPNC $\delta$-$\beta^*$ CM) are given. Moreover, this framework shows how core properties lift to the $m$-polar case and where new phenomena arise. Also examples and counterexamples are provided in order to separate the classes and to justify and illustrate the sharpness of the obtained results.