That is the first part of a series of studies on analyzing the higher-order teleparallel theory of gravity known as the f (T, B)-theory. This work attempts to understand how the anisotropic spacetimes are involved in the f (T, B)-theory. In this work, we review the previous analysis of f (T, B)-gravity, and we investigate the global dynamics in the case of a locally rotational Bianchi I background geometry. We focus in the case of f (T, B) = T + F (B) theory and we determine the criteria where the f (T, B)-theory solves the homogeneity problem. Finally, the integrability properties for the field equations are investigated by applying the Painlevé analysis.The analytic solution is expressed by a right Painlevé expansion.