There has been an increased interest in the variation diminishing properties of controlled linear time-invariant (LTI) systems and time-varying linear systems without inputs. In controlled LTI systems, these properties have recently been studied from the external perspective of -positive Hankel operators. Such systems have Hankel operators that diminish the number of sign changes (the variation) from past input to future output if the input variation is at most . For , this coincides with the classical class of externally positive systems. For linear systems without inputs, the focus has been on the internal perspective of -positive state-transition matrices, which diminish the variation of the initial system state. In the LTI case and for , this corresponds to the classical class of (unforced) positive systems. This paper bridges the gap between the internal and external perspectives of -positivity by analyzing internally Hankel -positive systems, which we define as state-space LTI systems where controllability and observability operators as well as the state-transition matrix are -positive. We show that the existing notions of external Hankel and internal -positivity are subsumed under internal Hankel -positivity, and we derive tractable conditions for verifying this property in the form of internal positivity of the first compound systems. As such, this class provides new means to verify external Hankel -positivity, and lays the foundation for future investigations of variation diminishing controlled linear systems. As an application, we use our framework to derive new bounds for the number of over- and undershoots in the step responses of LTI systems. Since our characterization defines a new positive realization problem, we also discuss geometric conditions for the existence of minimal internally Hankel -positive realizations.