Abstract 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 $\operatorname{GL}_{n}({{\mathbb{A}}})$ and its contragredient. As a consequence, we prove the trivialness of the relative period associated to a regular algebraic cuspidal automorphic representation of $\operatorname{GL}_{2n}({{\mathbb{A}}})$ of orthogonal type, which implies the algebraicity of the ratios of successive critical $L$-values for $\textrm{GSpin}_{2n}^{*} \times \operatorname{GL}_{n^{\prime}}$ by the result of Harder and Raghuram.