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The $q$-Onsager algebra $\mathcal O_q$ is defined by two generators $A$,$A^*$ and two relations, called the $q$-Dolan/Grady relations. Recently P.Baseilhac and S. Kolb found an automorphism $L$ of $\mathcal O_q$, that fixes$A$ and sends $A^*$ to a linear combination of $A^*$, $A^2A^*$, $AA^*A$,$A^*A^2$. Let $V$ denote an irreducible $\mathcal O_q$-module of finitedimension at least two, on which each of $A$, $A^*$ is diagonalizable. It isknown that $A$, $A^*$ act on $V$ as a tridiagonal pair of $q$-Racah type,giving access to four familiar elements $K$, $B$, $K^\downarrow$,$B^\downarrow$ in ${\rm End}(V)$ that are used to compare the eigenspacedecompositions for $A$, $A^*$ on $V$.  We display an invertible $H \in {\rm End}(V)$ such that $L(X)=H^{-1} X H$ on$V$ for all $X \in \mathcal O_q$. We describe what happens when one of $K$,$B$, $K^\downarrow$, $B^\downarrow$ is conjugated by $H$. For example$H^{-1}KH=a^{-1}A-a^{-2}K^{-1}$ where $a$ is a certain scalar that is used todescribe the eigenvalues of $A$ on $V$. We use the conjugation results tocompare the eigenspace decompositions for $A$, $A^*$, $L^{\pm 1}(A^*)$ on $V$.In this comparison we use the notion of an equitable triple; this is a 3-tupleof elements in ${\rm End}(V)$ such that any two satisfy a $q$-Weyl relation.Our comparison involves eight equitable triples. One of them is $a A - a^2 K$,$M^{-1}$, $K$ where $M= (a K-a^{-1} B)(a-a^{-1})^{-1}$. The map $M$ appears inearlier work of S. Bockting-Conrad concerning the double lowering operator$\psi$ of a tridiagonal pair.

Twisting finite-dimensional modules for the $q$-Onsager algebra $\mathcal O_q$ via the Lusztig automorphism