Symmetrical machines
A given machine has machines related to it by symmetry. They form Machine sets, which I also refer to as Multiplets.
3 relevant symmetries are Left-Right symmetry, Rewind and Reverse. It is also interesting to consider Goofy (or Inside-Outside symmetry as it applies e.g. to Inside Star to Outside Star). For these relevant symmetries, machines are classified as Singlets or as part of a Doublet.
In a general case, a given machine is part of a Left-Right Doublet, part of a Rewind Doublet, and part of a Reverse Doublet, such that there are 8 machines related by the 3 relevant symmetries, which I refer to as an Octuplet (16 if Goofy is applied to a pose, possibly more if Goofy can be applied separately to different poses).
Two machines in the same Left-Right Doublet are referred by the same name, so I find it useful to drop the Left-Right duplication and focus on the machine, its Rewind, its Reverse, and its Rewind Reverse. I refer to this as the machine Quadruplet.
The Rewind machine has the same poses as the machine. The Reverse machine and the Rewind Reverse machines have the Reverse poses.
This is the case for the 4 Step, which is part of the 4 Step set - a Quadruplet (it forms an Octuplet if you consider the Left-Right Doublets). Other cases can be found listed in machine sets.
In particular cases, machines can be more symmetrical. If the machine is invariant under the action of the respective symmetry, I refer to it as a Singlet of that symmetry. An example of this are Left-Right symmetrical machines, which I refer to as Left-Right Singlets. The I Infinity is the simplest Left-Right Singlet:
Beyond the 3 relevant symmetries, it is interesting to consider Goofy symmetry, which could be named Inside-Outside symmetry (as it applies to e.g. the relation between Inside Star and Outside Star). A machine can be identified as part of a Goofy Doublet or as a Goofy Singlet. For a 2-step machine with two poses, the Octuplet of 8 machines (with Left-Right being counted) can became a set of 16 if Goofy is applied to one of the poses, or even to 32 if Goofy is applied to both poses.
Taking the Ballerina and being strict, starting with:
1. Mono Throne L, Reverse Inside Star R
Apply Left-Right:
2. Mono Throne R, Reverse Inside Star L
Apply Rewind to 1, 2, gives 3, 4.
Apply Reverse to 1-4 gives 5-8 (5-8 have Mono Reverse Throne, Inside Star).
Apply Goofy to the Reverse Inside Star in 1-4, apply Goofy to the Inside Star in 5-8, you have 9-16 that I usually call Ballerouta.
Apply Goofy to the Mono Throne and Goofy to the Mono Reverse Throne, gives 17-32 that I usually call Goofy Ballerina and Goofy Ballerouta.
The last step in particular gives machines which are not that different from the respective machine in the 1-16 set, but strictly they are passing through different poses.