The equalities (Formula presented.) and (Formula presented.) are called the reverse order law and forward order law, respectively, for invertible matrices. However, these generally fail for singular/rectangular matrices, tensors of arbitrary order, and operators. Several researchers have worked in these problems starting with Greville [SIAM Rev. 8 (1966), 518–521: MR0210720]. Recently, Višnjić et al. [Electron. J. Linear Algebra 39 (2023), 379–394: MR4626656] studied the reverse order law for the (Formula presented.) -inverse in a unital ring. This manuscript establishes several sufficient conditions for the validity of both the reverse order law and forward order law for the new dual Moore–Penrose inverse (NDMPI) introduced in [J. Comput. Appl. Math., 454 (2025), 116185: MR4782122]. Additionally, some characterizations of the reverse order law of the NDMPI are obtained. We also explore a few possible theoretical applications of the reverse order law within this framework. Finally, we illustrate the additive property for dual complex matrices.