  We describe a new subclass of the class of real polynomials with real simple
roots called self-interlacing polynomials. This subclass is isomorphic to the
class of real Hurwitz stable polynomials (all roots in the open left
half-plane). In the work, we present basic properties of self-interlacing
polynomials and their relations with Hurwitz and Hankel matrices as well as
with Stiltjes type of continued fractions. We also establish "self-interlacing"
analogues of the well-known Hurwitz and Li\'enard-Chipart criterions for stable
polynomials. A criterion of Hurwitz stability of polynomials in terms of minors
of certain Hankel matrices is established.
