  In this paper we present a complete classification scheme for kinetic energy
operators (KEO's) describing a particle endowed with position-dependent mass
(PDM). We first present a generalized formulation of KEO's with PDM and show
that it is equivalent to three-parameter linear formulation. This reduces
further to two independent parameters under Hermiticity condition. Based on
this linear formulation we prove that, contrary to what was widely believed,
von Roos family is not the most general ordering. We found an entire new family
of Hermitian KEO's that does not fit into von Roos ordering. We were able to
construct all the Hermitian KEO's and classify them into two-parameter classes.
As one application, we solve the puzzling case of Yang and Yee KEO. We also
find, under certain conditions, some kind of duality between von Roos ordering
and one of the new classes.
