Every point of the domain of the bijective function generates only one point of the range, one-to-one, and every one-to-one function has an inverse function.
The even function is symmetric on the y-axis, such that f(-x) = f(x).
Therefore, the bijective function can't be even function. Is that conclusion right? and Is there any other relationship between the inverse and the symmetry property?
No even function from $\mathbb R$ into $\mathbb R$ is bijective because, for instance, $f(1)=f(-1)$. Almost the same argument shows that, if $a>0$, no even function from $(-a,a)$ into $\mathbb R$ is bijective (or, indeed, injective).
Note that some odd functions from $\mathbb R$ into $\mathbb R$ are not bijective too. Take, say, $f(x)=x^3-x$.