3-bit Walsh permutation/matrix columns

Cluster of the neighbor graph, on the right the matrix sums and vertex types

There are 25 transforms that look similar to the neutral position.
This is the case when the view from one or two axes remains the same or almost the same.
Almost the same means, that the original square is sheared into a (simple) parallelogram.

124

125 (as inverse)

136 (binary)

125 and 136 look similar to the neutral position.
125 looks the same from x and almost the same from y (here shown from −y).
136 looks almost the same from x.

These are the 25 permutations in the middle cluster of the positive component of the neighbor graph.
(There are only 18 transforms that do not look like a square or simple parallelogram from any side, namely those who's matrices have seven 1s.)

The big table below shows all 168 transforms in 28 rows and 6 columns.

The shown transforms are those with binary matrices. For the transforms with matrices that have some negative entries see here.

For the 25 rows with sums < 7 the position in the table corresponds with that in the neighbor graph: Left, middle or right cluster in the positive or negative component.

the other 3 rows

For the 3 rows with sum 7 the positions in the table correspond like this
to the positions in the small 3×3 matrix in the corner of each graph image:

positive
M	R	L
L	M	R
R	L	M

negative
R	L	M
L	M	R
M	R	L

conjugacy class	neut.	2+2			2+4		7a
cycle shape	neut.	Paris	Rome	Florence	Buenos Aires 5	Buenos Aires 6	Santiago +
sum	3	4	5a		5b	6	7
quantity	1	6	3	3	6	6	3
quantity	1	12			12		3

Permutations in the same row have the same complement pattern, and each complement pattern corresponds to four rows.

complement patterns to table rows

Each of the 28 table rows is identified by the vector (and corresponding matrix) in its main column.

weight 1

2+2
Rome

2+4
Buenos Aires 5

7a
Santiago +

1

724

526

364

753

2

174

165

534

673

4

127

136

325

657

weight 2

2+2
Paris

2+4
Buenos Aires 6

3

524

164

175

726

5

324

126

137

764

6

134

125

327

574

weight 3

neut.

2+2
Florence

7

124

135

326

564

The big table can be sorted by some properties of the permutations in the main column:

cc conjugacy class
cs cycle shape: abbreviated (city) name of the cycle shape; below the sum (3...7), which corresponds to the position in the cluster (which is why 5a and 5b are distinguished)
cp complement pattern: above the weight (1...3), below the value (1...7)
t triple of numbers used in the vectors in the row (number in ascending order)

Let $S$ be a permutation from the symmetric subgroup in the top row, and $M$ a permutation in the main column.
Then the permutation in the row of $M$ and the column of $S$ is $P=M\cdot S$ . E.g. 314 (second row, negative left column) is 134 ⋅ 214.