  Waldspurger's formula gives an identity between the norm of a torus period
and an L-function of the twist of an automorphic representation on GL(2). For
any two Hecke characters of a fixed quadratic extension, one can consider the
two torus periods coming from integrating one character against the automorphic
induction of the other. Because the corresponding L-functions agree, (the norms
of) these periods---which occur on different quaternion algebras---are closely
related. In this paper, we give a direct proof of an explicit identity between
the torus periods themselves.
