Regression on even function?

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Is there any test for whether or not the regression function is even?

Suppose we have a model: $Y=g(X, \epsilon)$, where $Y, X$ are both one dimensional. My questions is how do we test for $g$ is an even function or not? For example $Y=X^2+\epsilon$ (here $g(x)=x^2$, which is even on $x$).

Thanks.