  We explore the possibility that well known properties of the parity operator,
such as its idempotency and unitarity, might break down at the Planck scale.
Parity might then do more than just swap right and left polarized states and
reverse the sign of spatial momentum ${\bf k}$: it might generate
superpositions of right and left handed states, as well as mix momenta of
different magnitudes. We lay down the general formalism, but also consider the
concrete case of the Planck scale kinematics governed by $\kappa$-Poincar\'e
symmetries, where some of the general features highlighted appear explicitly.
We explore some of the observational implications for cosmological
fluctuations. Different power spectra for right handed and left handed tensor
modes might actually be a manifestation of deformed parity symmetry at the
Planck scale. Moreover, scale-invariance and parity symmetry appear deeply
interconnected.
