How to construct eigenvectors of angular momentum and directional parity (mirror)?

In the context of diatomic molecules I have seen it has come up that an eigenvalue $M_Lhbar$ of $L_z$ is doubly degenerate with respect to reflection about a plane containing the $z$ axis, whose corresponding operator I shall call $A_y$, since $A_yL_zA_y = -L_z$. Therefore one chooses the basis of $L_z^2,A_y$ instead of $L_z$ on its own since the hamiltonian also commutes with $A_y$.

My question is thus the following: how does one pass from the basis determined by $L_z^2,L_z$ to the basis determined by $L_z^2,A_y$?

$ |lambda,mrangle to |lambda,arangle ?$

where $lambda = |m|$ and $a$ is the eigenvalue ($pm1$) of $A_y$. This is a two dimensional space for which the two vectors $|lambda,mrangle,|lambda,-mrangle$ or $|lambda,+1rangle,|lambda,-1rangle$ form a basis.

Due to symmetry I expect something along the lines of:

$ |lambda,arangle = frac{1}{sqrt{2}} (|lambdarangle pm |-lambdarangle )$

where there is some link between $a,lambda$ and the sign of $pm$.

Since the real spherical harmonics $Y_{lm}$ are invariant under this mirror reflection I then thought that the answer would be like:

$ |lambda,pm1rangle = |lambdarangle pm (-1)^{lambda}|-lambdarangle $

But I really don’t know how to do this, I thought of using a similar method to the one used for spherical harmonics by means of the ladder operators but it doesn’t apply (at least in an obvious way) in this case. I’m pretty sure the answer is easy but I can’t figure it out.

Thanks for any help.