As every conditional expectation, $E[X\mid XY]=w(XY)$ for some measurable function $w$. Recall that:

Conditional expectations depend only on the joint distribution of the random variables considered, in the sense that, if $E[X\mid XY]=w(XY)$, then $E[X'\mid X'Y']=w(X'Y')$ for every $(X',Y')$ distributed like $(X,Y)$.

Choosing $(X',Y')=(-X,-Y)$ above, one gets $X'Y'=XY$ hence $$w(XY)=E[-X\mid XY]=-E[X\mid XY]=-w(XY).$$ Thus, $$E[X\mid XY]=0.
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One sees that $E[X\mid XY]=0$ for every *centered* gaussian vector $(X,Y)$, neither necessarily independent nor standard.

Still more generally:

Let $\Xi$ denote any centered gaussian vector, $u$ an *odd* measurable function such that $u(\Xi)$ is integrable and $v$ an *even* measurable function. Then, $$E[u(\Xi)\mid v(\Xi)]=0.$$