Integral of product of two periodic functions each with average zero

Question

Given two functions $f$ and $g$ which both have period $2\pi$ and average $0$ over the period, that is $f(x)=f(x+2\pi)$ and $\int_0^{2\pi} f(x) \rm{dx}=0$ (and the same for $g$) and also $f$ is known to be odd (or even, a shift in $x$ means it can be either). What can be said about the integral of their product over the period? $$\int_0^{2\pi} f(x)g(x)\rm{dx}=?$$ I'm hoping for this to be zero, so also - what relationships between the $f$ and $g$ would make this zero? ($f$ and $g$ can be assumed to be infinitely differentiable, finite, etc.)