A simplicial complex $K$ on a vertex set $[m] = \{1,...,m \}$ is self-dual if it is equal to its Alexander dual $\widehat{K}$, where $\widehat{K}$ is the simplicial complex on $[m]$ whose simplices are the complements of the missing faces of $K$. That is, simplices of $\widehat{K}$ are given by $$\widehat{K} = \{J \subseteq [m] \; | \; [m] \backslash J \notin K \}.$$

An example of a self dual complex is $\partial \Delta^2$ disjoint unioned with one vertex: $K = \{\emptyset, \{ 1\} \{2 \} \{ 3\} \{ 4\} \{ 1,2\} \{2,3 \} \{ 1,3\} \}$.

Timotejević's 2019 note explores the combinatorial structure of such complexes. For example, theorem 2.4 therein says that a complex $K$ is self dual if and only if for arbitrary $A \subseteq [m]$, $A \in K \iff [m] \backslash A \notin K$.

One may also define a simplicial complex to be self-dual if it is isomorphic, as a simplicial complex, to its Alexander dual. With this definition, the above theorem does not hold. For example, consider the simplicial complex on four vertices $$K = \{\emptyset, \{ 1\} \{2 \} \{ 3\} \{ 4\} \{ 1,2\} \{2,3 \} \{ 3,4\} \}.$$ This is isomorphic to its Alexander dual $\widehat{K} = \{\emptyset, \{ 1\} \{2 \} \{ 3\} \{ 4\} \{ 1,2\} \{1,3 \} \{ 2,4\} \}$, and is therefore self-dual in this sense.

What is known about self-dual complexes in the second sense? What is known about their combinatorial structure?