  For a positive integer $n$, a collection $S$ of subsets of
$[n]=\{1,\ldots,n\}$ is called 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 showed that the set of maximal
strongly (resp. weakly) separated collections in $2^{[n]}$ is connected via
flips, or mutations, ``in the presence of six (resp. four) witnesses''. We give
a symmetric analog of those results, by showing that each maximal symmetric
strongly (weakly) separated collection in $2^{[n]}$ can be obtained from any
other one by a series of special symmetric local transformations, so-called
symmetric flips. Also we establish the connectedness via symmetric flips for
the class of maximal symmetric $r$-separated collections in $2^{[n]}$ when
$n,r$ are 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 $B\setminus A$). This is related to a symmetric version of
higher Bruhat orders.
  These results are obtained as consequences of our study of related geometric
objects: symmetric rhombus and combined tilings and symmetric cubillages.
