= Solution
The <joint probability distribution> of all entries of $X$ is unchanged by $X\mapsto-X$, so $M$ and $-M$ have the same <probability distribution>. For the specified cyclic product $P=M_{i_1i_2}\cdots M_{i_ki_1}$, this transformation multiplies $P$ by $(-1)^k$. Every such product is an <integrable random variable>, since <normal random variables> have finite <moments> of all orders. Thus, for odd $k$,
$$
\mathbb EP=\mathbb E[(-1)^kP]=-\mathbb EP,
\qquad\boxed{\mathbb E[M_{i_1i_2}\cdots M_{i_ki_1}]=0.}
$$
Repeated indices cause no difficulty: all factors change sign simultaneously. It would be incorrect to assume that these factors are <independent>.
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