Here's a generalization of the result that eigenvectors of a real symmetric matrix with distinct eigenvalues are orthogonal:

Let $x$ and $y$ be unit eigenvalues of a (not necessarily symmetric) matrix $A$ with eigenvalues $\lambda$ and $\mu$, respectively.  Then
$$
\lambda\langle x,y\rangle
=\langle Ax,y\rangle
=\langle x,A^\mathrm{T}y\rangle
=\langle x,Ay\rangle-\langle x,(A-A^\mathrm{T})y\rangle
=\overline{\mu}\langle x,y\rangle-\langle x,(A-A^\mathrm{T})y\rangle.
$$
Rearranging and applying standard inequalities then yields
$$
|\langle x,y\rangle|
\leq\frac{\|A-A^\mathrm{T}\|_2}{|\lambda-\overline{\mu}|}.
$$
The numerator of this fraction makes intuitive sense, since any correlation between eigenvectors must come from the skew-symmetric part of the matrix.  Furthermore, the denominator provides the relationship you suggested in your question, especially when the eigenvalues are real.