Let's start from a system $f_n(x) = e^{inx}$.
I can rotate one element: $f_m(x + x_0) = e^{imx_0} e^{imx}$ , which is still orthogonal to all but one element of system.
Same holds for $n$-dimensional trigonometric systems on a torus: being shifted, a function is still orthogonal to all other elements.
When considering real-valued trigonometric systems, $\sin m(x + x_0)$ is orthogonal to all elements but $\sin m x$, $\cos m x$.
Question: do spherical functions enjoy same property? Say, if I rotate one of the spherical functions $Y^l_m(\theta, \phi)$, will it be orthogonal to all other functions (for the exception of finite subset?)
P.S. I've checked this for $l=0, l=1$.