  We consider self-adjoint Schr\"odinger operators in $L^2 (\mathbb{R}^d)$ with
a $\delta$-interaction of strength $\alpha$ and a $\delta'$-interaction of
strength $\beta$, respectively, supported on a hypersurface, where $\alpha$ and
$\beta^{-1}$ are bounded, real-valued functions. It is known that the
inequality $0 < \beta \leq 4/\alpha$ implies inequality of the eigenvalues of
these two operators below the bottoms of the essential spectra. We show that
this eigenvalue inequality is strict whenever $\beta < 4 / \alpha$ on a
nonempty, open subset of the hypersurface. Moreover, we point out special
geometries of the interaction support, such as broken lines or infinite cones,
for which strict inequality of the eigenvalues even holds in the borderline
case $\beta = 4 / \alpha$.
