Comparing Hilbert depth of $I$ with Hilbert depth of $S/I$

Let $I$ be a monomial ideal of $S=K[x_1,\ldots,x_n]$. We show that the following are equivalent: (i) $I$ is principal, (ii) $\operatorname{hdepth}(I)=n$, (iii) $\operatorname{hdepth}(S/I)=n-1$. Assuming that $I$ is squarefree, we prove that if $\operatorname{hdepth}(S/I)\leq 3$ or $n\leq 5$ then $\operatorname{hdepth}(I)\geq \operatorname{hdepth}(S/I)+1$. Also, we prove that if $\operatorname{hdepth}(S/I)\leq 5$ or $n\leq 7$ then then $\operatorname{hdepth}(I)\geq \operatorname{hdepth}(S/I)$.