Abstract: We study pairs of interpolators, the differentials they generate and their associated commutator theorems. An essential part of our analysis is the study of the intrinsic symmetries of the process. Since we work without any compatibility or categorical assumption, our results are flexible enough to obtain and generalize most known results for commutators or translation operators, in particular those of Cwikel et al. (Arkiv. Mat. 33:199-216, 1995) for differential methods and those of Carro et al. (Adv. Math. 169:241-312, 2002) for compatible pairs of interpolators. We also consider stability issues extending the results in Refs. (J. Inst. Math. Jussieu (to appear). http://arxiv.org/abs/1906.06677v1, 2020; Trans. Am. Math. Soc. 369:4671-4708, 2017; Adv. Comput. Math. 5(1006):329-359, 2007) from the complex method to general differential methods.

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