  A one-parameter family of operators that have the complementary Bannai-Ito
(CBI) polynomials as eigenfunctions is obtained. The CBI polynomials are the
kernel partners of the Bannai-Ito polynomials and also correspond to a
$q\rightarrow-1$ limit of the Askey-Wilson polynomials. The eigenvalue
equations for the CBI polynomials are found to involve second order Dunkl shift
operators with reflections and exhibit quadratic spectra. The algebra
associated to the CBI polynomials is given and seen to be a deformation of the
Askey-Wilson algebra with an involution. The relation between the CBI
polynomials and the recently discovered dual -1 Hahn and para-Krawtchouk
polynomials, as well as their relation with the symmetric Hahn polynomials, is
also discussed.
