3-bit Walsh permutation/seeds

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.
Each of their matrices has a different set of columns. (I.e., each of their vectors has entries from a different set of integers.)

There are 3*6=18 transform that do not look like a square or simple parallelogram from any side, namely those who's matrices have seven 1s.
They are also shown in three rows of the table below. The choice which to put in the left column is somewhat random.
The ones chosen are those not connected to the central cluster in the positive component. (I.e. the three remaining ones with black circles.)

The table shows some properties of the permutations in the left column:

conjugacy class	neut.	2+2			2+4
cycle shape		London	Rome	Florence	Lima 5	Lima 6	Rio +
sum	3	4	5a		5b	6	7
quantity	1	6	3	3	6	6	3
		12			12

Permutations in the same row have the same complement pattern:

2+2 Rome 2+4 Lima 5 7a Rio + 1 724 526 364 753 2 174 165 534 673 4 127 136 325 657

2+2 London 2+4 Lima 6 3 524 164 175 726 5 324 126 137 764 6 134 125 327 574

neut. 2+2 Florence 7 124 135 326 564

Each row contains the transform with the binary matrix above and that with an inverse matrix below.
(The latter is the same pattern of non-zero entries, but some 1s are negative.)
(The matrices for the first row are self-inverse.)

cc	cs	cp	t	stretch	stretched seeds
&0 neut.	&0 neut. 3	37	124	124	124	214	142	412	241	421
2+2	&1 London 4	26	134	134	134	314	143	413	341	431
2+2	&1 London 4	26	125	125	125	215	152	512	251	521
2+2	&1 London 4	25	234	324	324	234	342	432	243	423
2+2	&1 London 4	25	126	126	126	216	162	612	261	621
2+2	&1 London 4	23	245	524	524	254	542	452	245	425
2+2	&1 London 4	23	146	164	164	614	146	416	641	461
2+2	&2 Rome 5a	11	247	724	724	274	742	472	247	427
2+2	&2 Rome 5a	12	147	174	174	714	147	417	741	471
2+2	&2 Rome 5a	14	127	127	127	217	172	712	271	721
2+2	&3 Florence 5a	37	135	135	135	315	153	513	351	531
2+2	&3 Florence 5a	37	236	326	326	236	362	632	263	623
2+2	&3 Florence 5a	37	456	564	564	654	546	456	645	465
2+4	&4 Lima 5 5b	12	156	165	165	615	156	516	651	561
2+4	&4 Lima 5 5b	14	136	136	136	316	163	613	361	631
2+4	&4 Lima 5 5b	14	235	325	325	235	352	532	253	523
2+4	&4 Lima 5 5b	11	256	526	526	256	562	652	265	625
2+4	&4 Lima 5 5b	12	345	534	534	354	543	453	345	435
2+4	&4 Lima 5 5b	11	346	364	364	634	346	436	643	463
2+4	&5 Lima 6 6	23	157	175	175	715	157	517	751	571
2+4	&5 Lima 6 6	25	137	137	137	317	173	713	371	731
2+4	&5 Lima 6 6	23	267	726	726	276	762	672	267	627
2+4	&5 Lima 6 6	26	237	327	327	237	372	732	273	723
2+4	&5 Lima 6 6	25	467	764	764	674	746	476	647	467
2+4	&5 Lima 6 6	26	457	574	574	754	547	457	745	475
7a	&6 Rio +7	11	357	753	753	735	573	537	375	357
7a	&6 Rio +7	12	367	673	673	637	763	736	367	376
7a	&6 Rio +7	14	567	657	657	675	567	576	765	756