Consider the functions $x^2$ and $x^4 + 2x^2y^2$ on the unit sphere $S^2$. The surface integral of these functions over the sphere can easily be calculated by symmetry via $$3 \iint_{S^2} x^2 \mathrm{d}A = \iint_{S^2} (x^2 + y^2 + z^2) \, \mathrm{d}A = \iint_{S^2} \mathrm{d}A = 4\pi$$ and $$3 \iint_{S^2} (x^4+2x^2y^2)\, \mathrm{d}A = \iint_{S^2} (x^2 + y^2 + z^2)^2 \, \mathrm{d}A = \iint_{S^2} \mathrm{d}A = 4\pi.$$
However, I suspect (although I cannot prove) that the function $x^4$ cannot be integrated without direct parameterization of the sphere and evaluation of the surface integral.
My question is: in general, given any symmetries and polynomial relations on a manifold (in this case $(x, y, z) \mapsto (y, z, x)$ and $x^2 + y^2 + z^2 = 1$), is there a general theory to determine what functions are integrable over the manifold by symmetry and relations alone?
A reference (or definitive statement of lack thereof) would be greatly appreciated.