  Kochen-Specker contextuality is a fundamental feature of quantum mechanics
and a crucial resource for quantum computational advantage and reduction of
communication complexity. Its presence is witnessed in empirical data by the
violation of noncontextuality inequalities. However, all known noncontextuality
inequalities corresponding to facets of noncontextual polytopes are either Bell
inequalities or refer to cyclic or state-independent contextuality scenarios.
We introduce a general method for lifting noncontextuality inequalities,
deriving facets of noncontextual polytopes for more complex scenarios from
known facets of simpler subscenarios. Concretely, starting from an arbitrary
scenario, the addition of a new measurement or a new outcome preserves the
facet-defining nature of any noncontextuality inequality. This extends the
results of Pironio [J. Math. Phys. 46, 062112 (2005)] from Bell nonlocality
scenarios to contextuality scenarios, unifying liftings of Bell and
noncontextuality inequalities. Our method produces facet-defining
noncontextuality inequalities in all scenarios with contextual correlations,
and we present examples of facet-defining noncontextuality inequalities for
scenarios where no examples were known. Our results shed light on the structure
of noncontextuality polytopes and the relationship between such polytopes
across different scenarios.
