The generalization of the phenomena you see is genus theory. If $K = \mathbf{Q}(\sqrt{-d})$ and $H = K(j_d)$ then $H$ contains the Genus field $G$.
If $d = \prod_{i=1}^n p_i$ is squarefree (and odd for convenience's sake) then $G = K(\sqrt{p_i^*})$ where $p_i^* = (-1)^{(p-1)/2}$. In particular $p_i^* = p_i$ if and only if $p_i \equiv 1\bmod 4$. Therefore if $d \equiv 1\bmod 4$ then $d = \prod_i p_i^*$. Therefore if $d \equiv 1\bmod 4$ then $G$ and thus $H$ contains $K(\sqrt d) = K(i)$.
In general, if $d_1$ and $d_2$ have lots of prime factors in common, then the genus fields of their corresponding imaginary quadratic fields will also have large intersections. Outside of the genus field, I don't know of any studies into the intersections of Hilbert Class Fields.
The standard reference for this is Cox's wonderful book "Primes of the form $x^2 + ny^2$."