  The classes of externally Hankel $k$-positive LTI systems and autonomous
$k$-positive systems have recently been defined, and their properties and
applications began to be explored using the framework of total positivity and
variation diminishing operators. In this work, these two system classes are
subsumed under a new class of internally Hankel $k$-positive systems, which we
define as state-space LTI systems with $k$-positive controllability and
observability operators. We show that internal Hankel $k$-positivity is a
natural extension of the celebrated property of internal positivity ($k=1$),
and we derive tractable conditions for verifying the cases $k> 1$ in the form
of internal positivity of the first $k$ compound systems. As these conditions
define a new positive realization problem, we also discuss geometric conditions
for when a minimal internally Hankel $k$-positive realization exists. Finally,
we use our results to establish a new framework for bounding the number of
over- and undershoots in the step response of general LTI systems.
