  Topological phases realized in time-reversal invariant (TRI) systems are
foundational to experimental study of the broader canon of topological
condensed matter as they do not require exotic magnetic orders for realization.
We therefore introduce topological skyrmion phases of matter realized in TRI
systems as a foundational step towards experimental realization of topological
skyrmion phases. A novel bulk-boundary correspondence hidden from the ten-fold
way classification scheme is revealed by the presence of a non-trivial value of
a $\mathbb{Z}_2$ spin skyrmion invariant. This quantized topological invariant
gives a finer description of the topology in 2D TRI systems as it indicates the
presence or absence of robust helical edge states for open boundary conditions,
in cases where the $\mathbb{Z}_2$ invariant computed with projectors onto
occupied states takes a trivial value. Physically, we show this hidden
bulk-boundary correspondence derives from additional spin-momentum-locking of
the helical edge states associated with the topological skyrmion phase. ARPES
techniques and transport measurements can detect these signatures of
topological spin-momentum-locking and helical gapless modes. Our work therefore
lays the foundation for experimental study of these phases of matter.
