  We study a new family of "classical" orthogonal polynomials, here called big
-1 Jacobi polynomials, which satisfy (apart from a 3-term recurrence relation)
an eigenvalue problem with differential operators of Dunkl-type. These
polynomials can be obtained from the big $q$-Jacobi polynomials in the limit $q
\to -1$. An explicit expression of these polynomials in terms of Gauss'
hypergeometric functions is found. The big -1 Jacobi polynomials are orthogonal
on the union of two symmetric intervals of the real axis. We show that the big
-1 Jacobi polynomials can be obtained from the Bannai-Ito polynomials when the
orthogonality support is extended to an infinite number of points. We further
indicate that these polynomials provide a nontrivial realization of the
Askey-Wilson algebra for $q \to -1$.
