A reflection of a pattern evolves differently. C4 symmetry is preserved throughout.

Example: B36a*/S23 (clockwise symmetry)

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`x = 21, y = 12, rule = MAPARYXfhZofugWaH7oaIDogBZofuhogOiAaIHogIAAgAAWaH7oaIHogGiA6ICAAIAAaIDogIAAgACAAIAAAkAAAA`

3o5bo3b2o4b3o$2bo3bobo3bo5bobo$b2o3b3o3b3o3bo7$3o4b2o3bo5b3o$obo5bo3bo

bo3bo$2bo3b3o3b3o3b2o!

Ironically, only C1-symmetric non-totalistic neighbourhoods can allow this kind of C4 MAP rules.

------Henceforth, a bit of a digression

Moore neighbourhood is D4 symmetry (because the corners are not equivalent to edges)

Each isotropic non-totalistic neighbourhood, due to the rotate4reflect, can be analysed with basic group theory.

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`.....E.....2C4.....C2.....2σ(i).....2σ(n)`

A1...1.....1.....1.....1.....1

A2...1.....1.....1.....-1.....-1

B1...1.....-1....1.....1.....-1

B2...1.....-1....1.....-1.....1

E....2.....0.....-2.....0.....0

NB. the (i) and (n) for σ are used because that is the way we tend to refer to the Hensel notation of neighbourhoods. I will consider them equivalent to v(vertical) and d(dihedral) respectively.

Assigning symmetry to each neighbourhood should be easy (I have somehow messed up though and I think I have the wrong approach). Considering the 8 neighbours, what I have done that probably is wrong is to check each transformation and count the number of unchanged cells within the neighbourhood.

E=8; 2C4=0; C2=0; 2σ(i)=2; 2σ(n)=2

which is a reducible representation equivalent to 2 A1 + B1 + B2 + 2 E.

How do I move on and get the symmetries of each neighbourhood within this representation? It's obvious that for this thread title you need "chiral" neighbourhoods to get any kind of rotate-but-not-reflect thing going on.