It is shown that the theta functions of level $n$ on the principally polarised Prym varieties of an algebraic curve are dual to the sections of the orthogonal theta line bundle on the moduli space of Spin($n$)-bundles over the curve. As a by-product of our computations, we also note that when $n$ is odd, the Pfaffian line bundle on moduli space has a basis of sections labelled by the even theta characteristics of the curve.