  We analyze a family of singular Schr\"odinger operators with local singular
interactions supported by a hypersurface $\Sigma \subset \mathbb{R}^n, n \ge
2$, being the boundary of a Lipschitz domain, bounded or unbounded, not
necessarily connected. At each point of $\Sigma$ the interaction is
characterized by four real parameters, the earlier studied case of $\delta$-
and $\delta'$-interactions being particular cases. We discuss spectral
properties of these operators and derive operator inequalities between those
referring to the same hypersurface but different couplings and describe their
implications for spectral properties.
