All 3x3x3 involutions solved

I optimally solved all 170,911,549,184 involutions in half-turn metric. The entire process took about 2.5 weeks to generate the 37GB solution file. Source code is available on GitHub.

Here are the total and 48-way symmetry reduced counts by distance:

 d   Total          Unique by symmetry
--------------------------------------
 0   1              1
 1   6              1
 2   3              1
 3   72             2
 4   39             4
 5   960            25
 6   886            41
 7   12708          292
 8   19526          506
 9   160059         3501
10   349368         7741
11   2133822        45543
12   6903354        146698
13   33277628       700019
14   167193510      3500419
15   932736959      19478862
16   6251752591     130385528
17   37364321274    778842829
18   104805995145   2184417694
19   21346235247    445145591
20   456026         10842

Sum: 170911549184   3562686140

Solutions were found in four phases:

1. IDA* search to depth 18 for each of the 336,004 symmetry reduced edges-only involutions
2. Neighbor search (extend known solutions s by a move/antimove pair: m s m')
3. Twophase solver
4. Optimal solver for the remaining 6-7 million positions

The neighbor search was very effective in finding distance 19 solutions. That involutions can be extend in this manner is why I think the distance distribution is so top-heavy compared to all cubes.