  In this paper, under some regularity conditions, we prove a period relation
between the Betti--Whittaker periods associated to a regular algebraic cuspidal
automorphic representation of ${\rm GL}_n(\mathbb{A})$ and its contragredient.
As a consequence, we obtain the trivialness of the relative period associated
to a regular algebraic cuspidal automorphic representation of ${\rm
GL}_{2n}(\mathbb{A})$ of orthogonal type, which implies the algebraicity of the
ratios of successive critical $L$-values for ${\rm GSpin}_{2n}^* \times {\rm
GL}_{n'}$ by the result of Harder and Raghuram.
