W{\mathbb{P}}^n(q_0, \ldots, q_n)$ and $X_{A'} = Z(F_{A'}) \subseteq W{\mathbb{P}}^n(q_0', \ldots, q_n')$ and that $X_A$ and $X_{A'}$ are Calabi-Yau. Take $G$ and $G'$ to be subgroups of the group of phase symmetries that leave the respective cohomologies $H^{n,0}(X_A, {\mathbb{C}})$ and $H^{n,0}(X_{A'},{\mathbb{C}})$ to be invariant. We obtain two Calabi-Yau orbifolds $Z_{A,G}$ and $Z_{A', G'}$. One can find examples $Z_{A, G}$ and $Z_{A', G'}$ in the same family where their BHK mirrors $Z_{A^T, G^T}$ and $Z_{(A')^T, (G')^T}$ will be quotients of hypersurfaces in different weighted-projective spaces. See Section 5 for an explicit example.

Since the mirrors proposed by BHK and Batyrev-Borisov mirror symmetry are different, we ask the question of how we can relate them. Iritani suggested to look at the birational geometry of the mirrors $Z_{A^T, G^T}$ and $Z_{(A')^T, (G')^T}$. Shoemaker proved that when the hypersurfaces $X_A$ and $X_{A'}$ are in the same weighted-projective space and the groups $G$ and $G'$ are equal, then the BHK mirrors $Z_{A^T, G^T}$ and $Z_{(A')^T, (G')^T}$ are birational. He proves this claim by using a reinterpretation of BHK duality into toric language [@Sho]. In this paper, we drop one of these hypotheses and prove the following theorem.

\[main\] Let $Z_{A,G}$ and $Z_{A',G'}$ be Calabi-Yau orbifolds as 