A well known property of spherical harmonics is their orthogonality, which allows for nice decomposition of functions on the unit sphere.
The orthogonality of spherical harmonics takes the form of $$\int Y^*_{lm}(\theta,\phi) \, Y_{l'm'}(\theta,\phi) \sin\theta \,d\theta \, d\phi = \delta_{ll'} \delta_{mm'}$$
Generically speaking, this orthogonality only holds if you take an integral over the whole unit sphere, which makes complete sense given how the spherical harmonics are constructed in the first place.
However, upon visually inspecting some orbitals of the Hydrogen atom, I noticed that it appears as if the spherical harmonics are also orthogonal along (some) circular trajectories over the unit sphere. To check, I manually performed integrals over $\phi$ for $\int Y^*_{lm} Y_{lm'} \,d\phi$ for spherical harmonics at the same $l$ for fixed values of $\theta$. Curiously, these integrals were always zero, unless $m=m'$!
So my question is, are spherical harmonics of the same $l$ (but not $m$!) orthogonal on any (or some) circular trajectory $\tau$ over the unit sphere? $$\int Y^*_{lm} \, Y_{lm'}\, d\tau \stackrel{?}{=} \delta_{ll'} \delta_{mm'}$$ If not, then are they orthogonal on great circles (but not any generic circular trajectory)?
I have a feeling this pattern is accident that has to do my particular choice of $\theta$ and $Y_{lm}$'s that I checked, but maybe there is something more here? I know for a fact that the above relation is not true if we allow $l\neq l'$, but I am interested in the case where $l=l'$.
