I don't know much about cumulants. What I know is that they can be generated by using the cumulant generating function .
I'm studying the so-called Independent Component Analysis and my book says that if the pdfs involved are symmetric then the odd order cumulants are zero. Can you tell me why?

which cancel with the latter integral. Essentially, is an even function and is an odd function, therefore their product, the integrand is also an odd function which should integrate to 0 as expected.

The moment-generating function in this case can be expressed as

From the above result we know that the cumulant generating function is

As we know all the coefficients of odd powered terms in are 0, it should be the same for and thus ;
and by definition the cumulants satisfies

By comparing the coefficients, we can immediately claim that all the odd cumulants vanish.

There maybe some shorter and smarter ways to show this without going through the technical details in the above arguments. (There are always some technical conditions need to be considered whenever you are dealing with sum to infinity)