The q-Onsager algebra $${\mathcal {O}}_q$$ is defined by two generators A, $$A^*$$ and two relations, called the q-Dolan/Grady relations. Recently, Baseilhac and Kolb (Transform Groups, 2020, https://doi.org/10.1007/s00031-020-09555-7 ) 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 finite dimension at least two, on which each of A, $$A^*$$ is diagonalizable. It is known 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 $$\mathrm{End}(V)$$ that are used to compare the eigenspace decompositions for A, $$A^*$$ on V. We display an invertible $$H \in \mathrm{End}(V)$$ 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 to describe the eigenvalues of A on V. We use the conjugation results to compare 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-tuple of elements in $$\mathrm{End}(V)$$ 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 in earlier work of Bockting-Conrad (Linear Algebra Appl 437:242–270, 2012) concerning the double lowering operator $$\psi $$ of a tridiagonal pair.