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We study non-symmetric Jacobi polynomials of type $BC_{1}$ by means ofvector-valued and matrix-valued orthogonal polynomials. The interpretation asmatrix-valued orthogonal polynomials yields a new expression of thenon-symmetric Jacobi polynomials of type $BC_1$ in terms of the symmetricJacobi polynomials of type $BC_{1}$. In this interpretation, the Cherednikoperator, that has the non-symmetric Jacobi polynomials as eigenfunctions,corresponds to two shift operators for the symmetric Jacobi polynomials of type$BC_{1}$.  We show that the non-symmetric Jacobi polynomials of type $BC_{1}$ withso-called geometric root multiplicities, interpreted as vector-valuedpolynomials, can be identified with spherical functions on the sphere$S^{2m+1}=\mathrm{Spin}(2m+2)/\mathrm{Spin}(2m+1)$ associated with thefundamental spin-representation of $\mathrm{Spin}(2m+1)$. The Cherednikoperator corresponds to the Dirac operator for the spinors on $S^{2m+1}$ inthis interpretation.

Non-symmetric Jacobi polynomials of type $BC_{1}$ as vector-valued polynomials Part 1: spherical functions