Is any consistent first order extension of $\sf ZFC + CH$ interpretable in a consistent first order extension of $\sf ZFC + \neg CH$?
Is any consistent first order extension of $\sf ZFC + \neg CH$ interpretable in a consistent first order extension of $\sf ZFC + CH$?
Where $\sf CH$ is the continuum hypothesis.
If 1 is true and 2 is not, then $\sf ZFC + \neg CH$ is true!
If 2 is true and 1 is not, then $\sf ZFC + CH$ is true!
If both are true, then we have a dilemma?
If both are false, then the method is irrelevant?
Where we are?
We can get interpretations in both directions via a simple trick (using the obvious consistency assumptions).
Given a (consistent, computably axiomatizable) theory $\mathsf{A}$, consider the theory $\mathsf{ZFC+Con(A)}$; in any model $M$ of this theory by absoluteness we can pick out a canonical model of $\mathsf{A}$, namely the $L$-least model of $\mathsf{A}$ in the sense of $M$. This gives an interpretation of $\mathsf{A}$ in $\mathsf{ZFC+Con(A)}$, the shift to $L$ being used to make everything parameter-free.
Keep in mind that just because $X\in L$ does not mean $X\models \mathsf{V=L}$.
In particular, if $\mathsf{T}$ and $\mathsf{S}$ are extensions of $\mathsf{ZFC+CH}$ and $\mathsf{ZFC+\neg CH}$ respectively then $\mathsf{T}$ is interpretable in $\mathsf{ZFC+\neg CH+Con(T)}$ and $\mathsf{S}$ is interpretable in $\mathsf{ZFC+CH+Con(S)}$.
(None of this bears on the question of whether $\mathsf{CH}$ is true or not, however; there's no particular tension here.)