  New bispectral polynomials orthogonal on a Bannai-Ito bi-lattice (uniform
quadri-lattice) are obtained from an unconventional truncation of the
untruncated Bannai-Ito and complementary Bannai-Ito polynomials. A complete
characterization of the resulting para-Bannai-Ito polynomials is provided,
including a three term recurrence relation, a Dunkl-difference equation, an
explicit expression in terms of hypergeometric series and an orthogonality
relation. They are also derived as a $q\to -1$ limit of the $q$-para-Racah
polynomials. A connection to the dual $-1$ Hahn polynomials is also
established.
