= Solution
If either vector vanishes, both arguments are zero. Otherwise choose a <rotation matrix> $R$ sending $x/|x|$ to $y/|y|$. The <surface measure on a sphere> is rotation invariant. Changing variables $\sigma=R\omega$ gives
$$
\int_{\mathbb S^2}\phi(|x|y\cdot\sigma)\,d\sigma
=\int_{\mathbb S^2}\phi(|x||y|R(x/|x|)\cdot R\omega)\,d\omega
=\int_{\mathbb S^2}\phi(|y|x\cdot\omega)\,d\omega.
$$
Thus the two spherical <integrals> are equal. The property extends to complex continuous $\phi$ by treating real and imaginary parts separately, as needed for the Fourier exponential.
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