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Pure transposition rules....
[Perhaps symmetric]

In[]:=

Tuples[Range[3],2]

Out[]=

{{1,1},{1,2},{1,3},{2,1},{2,2},{2,3},{3,1},{3,2},{3,3}}

ParityPairings[Tuples[Range[0,2],2]]

Out[]=

{{{{0,0},{0,0}},{{0,1},{0,2}},{{1,0},{1,1}},{{1,2},{2,0}},{{2,1},{2,2}}},{{{0,0},{0,0}},{{0,1},{0,2}},{{1,0},{1,1}},{{1,2},{2,1}},{{2,0},{2,2}}},

⋯2616⋯

,{{{0,0},{2,2}},{{0,1},{0,1}},{{0,2},{0,2}},{{1,0},{1,0}},{{1,1},{1,1}},{{1,2},{1,2}},{{2,0},{2,0}},{{2,1},{2,1}}},{{{0,0},{0,0}},{{0,1},{0,1}},{{0,2},{0,2}},{{1,0},{1,0}},{{1,1},{1,1}},{{1,2},{1,2}},{{2,0},{2,0}},{{2,1},{2,1}},{{2,2},{2,2}}}}

Full expression not available

(

original memory size:

6.2 MB)

In[]:=

Take[%553,4]

Out[]=

{{{{0,0},{0,0}},{{0,1},{0,2}},{{1,0},{1,1}},{{1,2},{2,0}},{{2,1},{2,2}}},{{{0,0},{0,0}},{{0,1},{0,2}},{{1,0},{1,1}},{{1,2},{2,1}},{{2,0},{2,2}}},{{{0,0},{0,0}},{{0,1},{0,2}},{{1,0},{1,1}},{{1,2},{2,2}},{{2,0},{2,1}}},{{{0,0},{0,0}},{{0,1},{0,2}},{{1,0},{1,2}},{{1,1},{2,0}},{{2,1},{2,2}}}}

In[]:=

Rule@@@Union[Join[#,Reverse/@#]]&/@Take[%553,4]

Out[]=

{{{0,0}{0,0},{0,1}{0,2},{0,2}{0,1},{1,0}{1,1},{1,1}{1,0},{1,2}{2,0},{2,0}{1,2},{2,1}{2,2},{2,2}{2,1}},{{0,0}{0,0},{0,1}{0,2},{0,2}{0,1},{1,0}{1,1},{1,1}{1,0},{1,2}{2,1},{2,0}{2,2},{2,1}{1,2},{2,2}{2,0}},{{0,0}{0,0},{0,1}{0,2},{0,2}{0,1},{1,0}{1,1},{1,1}{1,0},{1,2}{2,2},{2,0}{2,1},{2,1}{2,0},{2,2}{1,2}},{{0,0}{0,0},{0,1}{0,2},{0,2}{0,1},{1,0}{1,2},{1,1}{2,0},{1,2}{1,0},{2,0}{1,1},{2,1}{2,2},{2,2}{2,1}}}

In[]:=

ArrayPlot[ResourceFunction["BlockCellularAutomaton"][#,CenterArray[Table[2,10],60],40],ColorRules->{0->White,1->Lighter[Orange],2->Darker[Orange]}]&/@(Rule@@@Union[Join[#,Reverse/@#]]&/@Take[%553,4])

Out[]=





In[]:=

symmetricQ[rules_]:=Sort[Map[Reverse,rules,{2}]]===Sort[rules]

In[]:=

Select[Rule@@@Union[Join[#,Reverse/@#]]&/@ParityPairings[Tuples[Range[0,2],2]],symmetricQ]

Out[]=

In[]:=

ResourceFunction["InteractiveListSelector"][(ArrayPlot[ResourceFunction["BlockCellularAutomaton"][#,CenterArray[Table[2,10],60],40],ColorRules->{0->White,1->Lighter[Orange],2->Darker[Orange]}]->#)&/@%559]

Out[]=



Add



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Non reversible