What is the intuitive reason behind vanishing of $a_n$ for odd $n$ in Fourier expansion of the full wave rectifier output signal?

Question

The full wave rectifier signal is defined by the following--- $$f(x)=|\sin(x)|$$ and has a Fourier series expansion of the form $$f(x)=\frac{2}{\pi}-\frac{4}{\pi}\sum_{n=0}^\infty \frac{\cos(2nx)}{4n^2-1}$$ which is easy to obtain directly by formulas for the Fourier coefficients. It is seen that the $a_n$s with odd $n$ vanish identically. What is the intuitive reason for this? I mean, is there any way to guess without doing the integrations and steps involved in calculating $a_n$ that only $a_n$ with even $n$ will remain?