  For a positive integer $n$, a collection $S$ of subsets of
$[n]=\{1,\ldots,n\}$ is called \emph{symmetric} if $X\in S$ implies $X^\ast\in
S$, where $X^\ast:=\{i\in [n]\colon n-i+1\notin X\}$ (the involution $\ast$ was
introduced by Karpman). Leclerc and Zelevinsky established the connectedness of
the class of maximal strongly separated collections in $2^{[n]}$ via flips
(mutations) "in the presence of six witnesses". As a counterpart of that
result, we show that any two maximal symmetric strongly separated collections
in $2^{[n]}$ are connected by a series of special symmetric local
transformations, so-called \emph{symmetric flips}. Next we obtain similar
results on connectedness of the class of maximal symmetric weakly separated
collections in $2^{[n]}$, and the class of maximal symmetric $r$-separated
collections in $2^{[n]}$ when $n$ is even (where sets $A,B\subseteq [n]$ are
called $r$-separated if there are no elements $i_0<i_1< \cdots <i_{r+1}$ in
$[n]$ which alternate in $A\setminus B$ and $B\setminus A$).
  These results are obtained as consequences of our study of related geometric
objects: symmetric rhombus and combined tilings and symmetric cubillages.
