For a lattice \(L\), let \(\text{Sub}(L)\) denote the lattice of all sublattices of \(L\), and \(L^d\) the dual of \(L\). If \(L\) and \(K\) are lattices and \(L\approx K\) or \(L\approx K^d\), then \(\text{Sub}(L)\approx \text{Sub}(K)\). The converse in not true in general. The scope of this paper is to prove that if \(L\) and \(K\) are lattices with \(\text{Sub}(L)\approx \text{Sub}(K)\), then: i) If \(L\) is simple then \(L\approx K\) or \(L\approx K^d\). ii) If \(L\) satisfies a self-dual infinitary Horn sentence \(\psi\) stronger than the modular identity, then \(\psi\) holds in \(K\). A corollary of ii) is: A modular lattice variety is closed under isomorphisms of sublattice-lattices iff it is self-dual.